The FREQ Procedure (Book Excerpt)

Transcription

1 SAS/STAT 9.22 User s Guide The FREQ Procedure (Book Excerpt) SAS Documentation

2 This document is an individual chapter from SAS/STAT 9.22 User s Guide. The correct bibliographic citation for the complete manual is as follows: SAS Institute Inc SAS/STAT 9.22 User s Guide. Cary, NC: SAS Institute Inc. Copyright 2010, SAS Institute Inc., Cary, NC, USA All rights reserved. Produced in the United States of America. For a Web download or e-book: Your use of this publication shall be governed by the terms established by the vendor at the time you acquire this publication. U.S. Government Restricted Rights Notice: Use, duplication, or disclosure of this software and related documentation by the U.S. government is subject to the Agreement with SAS Institute and the restrictions set forth in FAR , Commercial Computer Software-Restricted Rights (June 1987). SAS Institute Inc., SAS Campus Drive, Cary, North Carolina st electronic book, May 2010 SAS Publishing provides a complete selection of books and electronic products to help customers use SAS software to its fullest potential. For more information about our e-books, e-learning products, CDs, and hard-copy books, visit the SAS Publishing Web site at support.sas.com/publishing or call SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in the USA and other countries. indicates USA registration. Other brand and product names are registered trademarks or trademarks of their respective companies.

3 Chapter 35 The FREQ Procedure Contents Overview: FREQ Procedure Getting Started: FREQ Procedure Frequency Tables and Statistics Agreement Study Syntax: FREQ Procedure PROC FREQ Statement BY Statement EXACT Statement OUTPUT Statement TABLES Statement TEST Statement WEIGHT Statement Details: FREQ Procedure Inputting Frequency Counts Grouping with Formats Missing Values In-Database Computation Statistical Computations Definitions and Notation Chi-Square Tests and Statistics Measures of Association Binomial Proportion Risks and Risk Differences Odds Ratio and Relative Risks for 2 x 2 Tables Cochran-Armitage Test for Trend Jonckheere-Terpstra Test Tests and Measures of Agreement Cochran-Mantel-Haenszel Statistics Gail-Simon Test for Qualitative Interactions Exact Statistics Computational Resources Output Data Sets Displayed Output ODS Table Names

4 2210 Chapter 35: The FREQ Procedure ODS Graphics Examples: FREQ Procedure Example 35.1: Output Data Set of Frequencies Example 35.2: Frequency Dot Plots Example 35.3: Chi-Square Goodness-of-Fit Tests Example 35.4: Binomial Proportions Example 35.5: Analysis of a 2x2 Contingency Table Example 35.6: Output Data Set of Chi-Square Statistics Example 35.7: Cochran-Mantel-Haenszel Statistics Example 35.8: Cochran-Armitage Trend Test Example 35.9: Friedman s Chi-Square Test Example 35.10: Cochran s Q Test References Overview: FREQ Procedure The FREQ procedure produces one-way to n-way frequency and contingency (crosstabulation) tables. For two-way tables, PROC FREQ computes tests and measures of association. For n-way tables, PROC FREQ provides stratified analysis by computing statistics across, as well as within, strata. For one-way frequency tables, PROC FREQ computes goodness-of-fit tests for equal proportions or specified null proportions. For one-way tables, PROC FREQ also provides confidence limits and tests for binomial proportions, including tests for noninferiority and equivalence. For contingency tables, PROC FREQ can compute various statistics to examine the relationships between two classification variables. For some pairs of variables, you might want to examine the existence or strength of any association between the variables. To determine if an association exists, chi-square tests are computed. To estimate the strength of an association, PROC FREQ computes measures of association that tend to be close to zero when there is no association and close to the maximum (or minimum) value when there is perfect association. The statistics for contingency tables include the following: chi-square tests and measures measures of association risks (binomial proportions) and risk differences for 2 2 tables odds ratios and relative risks for 2 2 tables tests for trend tests and measures of agreement Cochran-Mantel-Haenszel statistics

5 Overview: FREQ Procedure 2211 PROC FREQ computes asymptotic standard errors, confidence intervals, and tests for measures of association and measures of agreement. Exact p-values and confidence intervals are available for many test statistics and measures. PROC FREQ also performs analyses that adjust for any stratification variables by computing statistics across, as well as within, strata for n-way tables. These statistics include Cochran-Mantel-Haenszel statistics and measures of agreement. In choosing measures of association to use in analyzing a two-way table, you should consider the study design (which indicates whether the row and column variables are dependent or independent), the measurement scale of the variables (nominal, ordinal, or interval), the type of association that each measure is designed to detect, and any assumptions required for valid interpretation of a measure. You should exercise care in selecting measures that are appropriate for your data. Similar comments apply to the choice and interpretation of test statistics. For example, the Mantel- Haenszel chi-square statistic requires an ordinal scale for both variables and is designed to detect a linear association. The Pearson chi-square, on the other hand, is appropriate for all variables and can detect any kind of association, but it is less powerful for detecting a linear association because its power is dispersed over a greater number of degrees of freedom (except for 2 2 tables). For more information about selecting the appropriate statistical analyses, see Agresti (2007) or Stokes, Davis, and Koch (2000). Several SAS procedures produce frequency counts; only PROC FREQ computes chi-square tests for one-way to n-way tables and measures of association and agreement for contingency tables. Other procedures to consider for counting include the TABULATE and UNIVARIATE procedures. When you want to produce contingency tables and tests of association for sample survey data, use PROC SURVEYFREQ. See Chapter 14, Introduction to Survey Procedures, for more information. When you want to fit models to categorical data, use a procedure such as CATMOD, GENMOD, GLIMMIX, LOGISTIC, PROBIT, or SURVEYLOGISTIC. See Chapter 8, Introduction to Categorical Data Analysis Procedures, for more information. PROC FREQ uses the Output Delivery System (ODS), a SAS subsystem that provides capabilities for displaying and controlling the output from SAS procedures. ODS enables you to convert any of the output from PROC FREQ into a SAS data set. See the section ODS Table Names on page 2331 for more information. PROC FREQ now uses ODS Graphics to create graphs as part of its output. For general information about ODS Graphics, see Chapter 21, Statistical Graphics Using ODS. For specific information about the statistical graphics available with the FREQ procedure, see the PLOTS option in the TABLES statement and the section ODS Graphics on page 2334.

6 2212 Chapter 35: The FREQ Procedure Getting Started: FREQ Procedure Frequency Tables and Statistics The FREQ procedure provides easy access to statistics for testing for association in a crosstabulation table. In this example, high school students applied for courses in a summer enrichment program; these courses included journalism, art history, statistics, graphic arts, and computer programming. The students accepted were randomly assigned to classes with and without internships in local companies. Table 35.1 contains counts of the students who enrolled in the summer program by gender and whether they were assigned an internship slot. Table 35.1 Summer Enrichment Data Enrollment Gender Internship Yes No Total boys yes boys no girls yes girls no The SAS data set SummerSchool is created by inputting the summer enrichment data as cell count data, or providing the frequency count for each combination of variable values. The following DATA step statements create the SAS data set SummerSchool: data SummerSchool; input Gender $ Internship $ Enrollment $ datalines; boys yes yes 35 boys yes no 29 boys no yes 14 boys no no 27 girls yes yes 32 girls yes no 10 girls no yes 53 girls no no 23 ; The variable Gender takes the values boys or girls, the variable Internship takes the values yes and no, and the variable Enrollment takes the values yes and no. The variable Count contains the number of students that correspond to each combination of data values. The double at sign (@@) indicates that more than one observation is included on a single data line. In this DATA step, two observations are included on each line. Researchers are interested in whether there is an association between internship status and summer program enrollment. The Pearson chi-square statistic is an appropriate statistic to assess the association in the corresponding 2 2 table. The following PROC FREQ statements specify this analysis.

7 Frequency Tables and Statistics 2213 You specify the table for which you want to compute statistics with the TABLES statement. You specify the statistics you want to compute with options after a slash (/) in the TABLES statement. proc freq data=summerschool order=data; tables Internship*Enrollment / chisq; weight Count; run; The ORDER= option controls the order in which variable values are displayed in the rows and columns of the table. By default, the values are arranged according to the alphanumeric order of their unformatted values. If you specify ORDER=DATA, the data are displayed in the same order as they occur in the input data set. Here, because yes appears before no in the data, yes appears first in any table. Other options for controlling order include ORDER=FORMATTED, which orders according to the formatted values, and ORDER=FREQUENCY, which orders by descending frequency count. In the TABLES statement, Internship*Enrollment specifies a table where the rows are internship status and the columns are program enrollment. The CHISQ option requests chi-square statistics for assessing association between these two variables. Because the input data are in cell count form, the WEIGHT statement is required. The WEIGHT statement names the variable Count, which provides the frequency of each combination of data values. Figure 35.1 presents the crosstabulation of Internship and Enrollment. In each cell, the values printed under the cell count are the table percentage, row percentage, and column percentage, respectively. For example, in the first cell, percent of the students offered courses with internships accepted them and percent did not. Figure 35.1 Crosstabulation Table The FREQ Procedure Table of Internship by Enrollment Internship Enrollment Frequency Percent Row Pct Col Pct yes no Total yes no Total

8 2214 Chapter 35: The FREQ Procedure Figure 35.2 displays the statistics produced by the CHISQ option. The Pearson chi-square statistic is labeled Chi-Square and has a value of with 1 degree of freedom. The associated p-value is , which means that there is no significant evidence of an association between internship status and program enrollment. The other chi-square statistics have similar values and are asymptotically equivalent. The other statistics (phi coefficient, contingency coefficient, and Cramer s V ) are measures of association derived from the Pearson chi-square. For Fisher s exact test, the two-sided p-value is , which also shows no association between internship status and program enrollment. Figure 35.2 Statistics Produced with the CHISQ Option Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Continuity Adj. Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V Fisher's Exact Test Cell (1,1) Frequency (F) 67 Left-sided Pr <= F Right-sided Pr >= F Table Probability (P) Two-sided Pr <= P The analysis, so far, has ignored gender. However, it might be of interest to ask whether program enrollment is associated with internship status after adjusting for gender. You can address this question by doing an analysis of a set of tables (in this case, by analyzing the set consisting of one for boys and one for girls). The Cochran-Mantel-Haenszel (CMH) statistic is appropriate for this situation: it addresses whether rows and columns are associated after controlling for the stratification variable. In this case, you would be stratifying by gender. The PROC FREQ statements for this analysis are very similar to those for the first analysis, except that there is a third variable, Gender, in the TABLES statement. When you cross more than two variables, the two rightmost variables construct the rows and columns of the table, respectively, and the leftmost variables determine the stratification. The following PROC FREQ statements also request frequency plots for the crosstabulation tables. PROC FREQ produces these plots by using ODS Graphics to create graphs as part of the procedure output. Before requesting graphs, you must enable ODS Graphics with the ODS GRAPHICS ON statement.

9 Frequency Tables and Statistics 2215 ods graphics on; proc freq data=summerschool; tables Gender*Internship*Enrollment / chisq cmh plots(only)=freqplot; weight Count; run; ods graphics off; This execution of PROC FREQ first produces two individual crosstabulation tables of Internship by Enrollment: one for boys and one for girls. Frequency plots and chi-square statistics are produced for each individual table. Figure 35.3, Figure 35.4, and Figure 35.5 show the results for boys. Note that the chi-square statistic for boys is significant at the D 0:05 level of significance. Boys offered a course with an internship are more likely to enroll than boys who are not. Figure 35.4 displays the frequency plot of Internship by Enrollment for boys. By default, the frequency plot is displayed as a bar chart with vertical grouping by the row variable Internship. You can use PLOTS= options to request a dot plot instead of a bar chart or to change the orientation of the bars from vertical to horizontal. You can also use PLOTS= options to specify other two-way layouts such as stacked or horizontal grouping. Figure 35.6, Figure 35.7, and Figure 35.8 display the crosstabulation table, frequency plot, and chi-square statistics for girls. You can see that there is no evidence of association between internship offers and program enrollment for girls. Figure 35.3 Crosstabulation Table for Boys The FREQ Procedure Table 1 of Internship by Enrollment Controlling for Gender=boys Internship Enrollment Frequency Percent Row Pct Col Pct no yes Total no yes Total

10 2216 Chapter 35: The FREQ Procedure Figure 35.4 Frequency Plot for Boys Figure 35.5 Chi-Square Statistics for Boys Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Continuity Adj. Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V Fisher's Exact Test Cell (1,1) Frequency (F) 27 Left-sided Pr <= F Right-sided Pr >= F Table Probability (P) Two-sided Pr <= P

11 Frequency Tables and Statistics 2217 Figure 35.6 Crosstabulation Table for Girls Table 2 of Internship by Enrollment Controlling for Gender=girls Internship Enrollment Frequency Percent Row Pct Col Pct no yes Total no yes Total Figure 35.7 Frequency Plot for Girls

12 2218 Chapter 35: The FREQ Procedure Figure 35.8 Chi-Square Statistics for Girls Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Continuity Adj. Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V Fisher's Exact Test Cell (1,1) Frequency (F) 23 Left-sided Pr <= F Right-sided Pr >= F Table Probability (P) Two-sided Pr <= P These individual table results demonstrate the occasional problems with combining information into one table and not accounting for information in other variables such as Gender. Figure 35.9 contains the CMH results. There are three summary (CMH) statistics; which one you use depends on whether your rows and/or columns have an order in r c tables. However, in the case of 2 2 tables, ordering does not matter and all three statistics take the same value. The CMH statistic follows the chi-square distribution under the hypothesis of no association, and here, it takes the value with 1 degree of freedom. The associated p-value is , which indicates a significant association at the D 0:05 level. Thus, when you adjust for the effect of gender in these data, there is an association between internship and program enrollment. But, if you ignore gender, no association is found. Note that the CMH option also produces other statistics, including estimates and confidence limits for relative risk and odds ratios for 2 2 tables and the Breslow-Day Test. These results are not displayed here. Figure 35.9 Test for the Hypothesis of No Association Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Prob Nonzero Correlation Row Mean Scores Differ General Association

13 Agreement Study 2219 Agreement Study Medical researchers are interested in evaluating the efficacy of a new treatment for a skin condition. Dermatologists from participating clinics were trained to conduct the study and to evaluate the condition. After the training, two dermatologists examined patients with the skin condition from a pilot study and rated the same patients. The possible evaluations are terrible, poor, marginal, and clear. Table 35.2 contains the data. Table 35.2 Skin Condition Data Dermatologist 2 Dermatologist 1 Terrible Poor Marginal Clear Terrible Poor Marginal Clear The following DATA step statements create the SAS dataset SkinCondition. The dermatologists evaluations of the patients are contained in the variables Derm1 and Derm2; the variable Count is the number of patients given a particular pair of ratings. data SkinCondition; input Derm1 $ Derm2 $ Count; datalines; terrible terrible 10 terrible poor 4 terrible marginal 1 terrible clear 0 poor terrible 5 poor poor 10 poor marginal 12 poor clear 2 marginal terrible 2 marginal poor 4 marginal marginal 12 marginal clear 5 clear terrible 0 clear poor 2 clear marginal 6 clear clear 13 ; The following PROC FREQ statements request an agreement analysis of the skin condition data. In order to evaluate the agreement of the diagnoses (a possible contribution to measurement error in the study), the kappa coefficient is computed. The AGREE option in the TABLES statement requests the kappa coefficient, together with its standard error and confidence limits. The KAPPA option in the TEST statement requests a test for the null hypothesis that kappa equals zero, or that the agreement is purely by chance.

14 2220 Chapter 35: The FREQ Procedure proc freq data=skincondition order=data; tables Derm1*Derm2 / agree noprint; test kappa; weight Count; run; Figure shows the results. The kappa coefficient has the value , which indicates slight agreement between the dermatologists, and the hypothesis test confirms that you can reject the null hypothesis of no agreement. This conclusion is further supported by the confidence interval of (0.2030, ), which suggests that the true kappa is greater than zero. The AGREE option also produces Bowker s test for symmetry and the weighted kappa coefficient, but that output is not shown here. Figure Agreement Study The FREQ Procedure Statistics for Table of Derm1 by Derm2 Simple Kappa Coefficient Kappa ASE % Lower Conf Limit % Upper Conf Limit Test of H0: Kappa = 0 ASE under H Z One-sided Pr > Z <.0001 Two-sided Pr > Z <.0001 Syntax: FREQ Procedure The following statements are available in PROC FREQ: PROC FREQ < options > ; BY variables ; EXACT statistic-options < / computation-options > ; OUTPUT < OUT=SAS-data-set > options ; TABLES requests < / options > ; TEST options ; WEIGHT variable < / option > ; The PROC FREQ statement is the only required statement for the FREQ procedure. If you specify the following statements, PROC FREQ produces a one-way frequency table for each variable in the most recently created data set. proc freq; run;

15 PROC FREQ Statement 2221 The rest of this section gives detailed syntax information for the BY, EXACT, OUTPUT, TABLES, TEST, and WEIGHT statements in alphabetical order after the description of the PROC FREQ statement. Table 35.3 summarizes the basic function of each PROC FREQ statement. Table 35.3 Summary of PROC FREQ Statements Statement Description BY Provides separate analyses for each BY group EXACT Requests exact tests OUTPUT Requests an output data set TABLES Specifies tables and requests analyses TEST Requests tests for measures of association and agreement WEIGHT Identifies a weight variable PROC FREQ Statement PROC FREQ < options > ; The PROC FREQ statement invokes the procedure and optionally identifies the input data set. By default, the procedure uses the most recently created SAS data set. Table 35.4 lists the options available in the PROC FREQ statement. Descriptions follow in alphabetical order. Table 35.4 PROC FREQ Statement Options Option Description COMPRESS Begins the next one-way table on the current page DATA= Names the input data set FORMCHAR= Specifies the outline and cell divider characters for crosstabulation tables NLEVELS Displays the number of levels for all TABLES variables NOPRINT Suppresses all displayed output ORDER= Specifies the order for reporting variable values PAGE Displays one table per page You can specify the following options in the PROC FREQ statement. COMPRESS begins display of the next one-way frequency table on the same page as the preceding one-way table if there is enough space to begin the table. By default, the next one-way table begins on the current page only if the entire table fits on that page. The COMPRESS option is not valid with the PAGE option. DATA=SAS-data-set names the SAS data set to be analyzed by PROC FREQ. If you omit the DATA= option, the procedure uses the most recently created SAS data set.

16 2222 Chapter 35: The FREQ Procedure FORMCHAR(1,2,7)= formchar-string defines the characters to be used for constructing the outlines and dividers for the cells of crosstabulation table displays. The formchar-string should be three characters long. The characters are used to draw the vertical separators (1), the horizontal separators (2), and the vertical-horizontal intersections (7). If you do not specify the FORMCHAR= option, PROC FREQ uses FORMCHAR(1,2,7)= -+ by default. Table 35.5 summarizes the formatting characters used by PROC FREQ. Table 35.5 Formatting Characters Used by PROC FREQ Position Default Used to Draw 1 Vertical separators 2 - Horizontal separators 7 + Intersections of vertical and horizontal separators The FORMCHAR= option can specify 20 different SAS formatting characters used to display output; however, PROC FREQ uses only the first, second, and seventh formatting characters. Therefore, the proper specification for PROC FREQ is FORMCHAR(1,2,7)= formcharstring. Specifying all blanks for formchar-string produces crosstabulation tables with no outlines or dividers for example, FORMCHAR(1,2,7)=. You can use any character in formchar-string, including hexadecimal characters. If you use hexadecimal characters, you must put an x after the closing quote. For information about which hexadecimal codes to use for which characters, see the documentation for your hardware. See the CALENDAR, PLOT, and TABULATE procedures in the Base SAS Procedures Guide for more information about form characters. NLEVELS displays the Number of Variable Levels table, which provides the number of levels for each variable named in the TABLES statements. See the section Number of Variable Levels Table on page 2322 for details. PROC FREQ determines the variable levels from the formatted variable values, as described in the section Grouping with Formats on page NOPRINT suppresses the display of all output. You can use the NOPRINT option when you only want to create an output data set. See the section Output Data Sets on page 2320 for information about the output data sets produced by PROC FREQ. Note that the NOPRINT option temporarily disables the Output Delivery System (ODS). For more information, see Chapter 20, Using the Output Delivery System. NOTE: A NOPRINT option is also available in the TABLES statement. It suppresses display of the crosstabulation tables but allows display of the requested statistics. ORDER=DATA FORMATTED FREQ INTERNAL specifies the order of the variable levels in the frequency and crosstabulation tables, which you request in the TABLES statement.

17 BY Statement 2223 The ORDER= option can take the following values: Value of ORDER= DATA FORMATTED FREQ INTERNAL Levels Ordered By Order of appearance in the input data set External formatted value, except for numeric variables with no explicit format, which are sorted by their unformatted (internal) value Descending frequency count; levels with the most observations come first in the order Unformatted value By default, ORDER=INTERNAL. The FORMATTED and INTERNAL orders are machinedependent. The ORDER= option does not apply to missing values, which are always ordered first. For more information about sorting order, see the chapter on the SORT procedure in the Base SAS Procedures Guide and the discussion of BY-group processing in SAS Language Reference: Concepts. PAGE displays only one table per page. Otherwise, PROC FREQ displays multiple tables per page as space permits. The PAGE option is not valid with the COMPRESS option. BY Statement BY variables ; You can specify a BY statement with PROC FREQ to obtain separate analyses on observations in groups defined by the BY variables. When a BY statement appears, the procedure expects the input data set to be sorted in order of the BY variables. If your input data set is not sorted in ascending order, use one of the following alternatives: Sort the data by using the SORT procedure with a similar BY statement. Specify the BY statement option NOTSORTED or DESCENDING in the BY statement for the FREQ procedure. The NOTSORTED option does not mean that the data are unsorted but rather that the data are arranged in groups (according to values of the BY variables) and that these groups are not necessarily in alphabetical or increasing numeric order. Create an index on the BY variables by using the DATASETS procedure. For more information about the BY statement, see SAS Language Reference: Concepts. For more information about the DATASETS procedure, see the Base SAS Procedures Guide.

18 2224 Chapter 35: The FREQ Procedure EXACT Statement EXACT statistic-options < / computation-options > ; The EXACT statement requests exact tests or confidence limits for the specified statistics. Optionally, PROC FREQ computes Monte Carlo estimates of the exact p-values. The statistic-options specify the statistics to provide exact tests or confidence limits for. The computation-options specify options for the computation of exact statistics. See the section Exact Statistics on page 2314 for details. NOTE: PROC FREQ computes exact tests with fast and efficient algorithms that are superior to direct enumeration. Exact tests are appropriate when a data set is small, sparse, skewed, or heavily tied. For some large problems, computation of exact tests might require a considerable amount of time and memory. Consider using asymptotic tests for such problems. Alternatively, when asymptotic methods might not be sufficient for such large problems, consider using Monte Carlo estimation of exact p-values. See the section Computational Resources on page 2317 for more information. Statistic-Options The statistic-options specify the statistics to provide exact tests or confidence limits for. For one-way tables, exact p-values are available for the binomial proportion tests and the chi-square goodness-of-fit test. Exact confidence limits are available for the binomial proportion. For two-way tables, exact p-values are available for the following tests: Pearson chi-square test, likelihood-ratio chi-square test, Mantel-Haenszel chi-square test, Fisher s exact test, Jonckheere- Terpstra test, and Cochran-Armitage test for trend. Exact p-values are also available for tests of the following statistics: Pearson correlation coefficient, Spearman correlation coefficient, Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, Somers D.RjC /, simple kappa coefficient, and weighted kappa coefficient. For 2 2 tables, PROC FREQ provides exact confidence limits for the odds ratio, exact unconditional confidence limits for the proportion difference, and McNemar s exact test. For stratified 2 2 tables, PROC FREQ provides Zelen s exact test for equal odds ratios, exact confidence limits for the common odds ratio, and an exact test for the common odds ratio. Table 35.6 lists the available statistic-options and the exact statistics computed. Most of the option names are identical to the corresponding option names in the TABLES and OUTPUT statements. You can request exact computations for groups of statistics by using options that are identical to the following TABLES statement options: CHISQ, MEASURES, and AGREE. For example, when you specify the CHISQ option in the EXACT statement, PROC FREQ computes exact p-values for the Pearson chi-square, likelihood-ratio chi-square, and Mantel-Haenszel chi-square tests. You can request exact computations for an individual statistic by specifying the corresponding statistic-option from the list in Table 35.6.

19 EXACT Statement 2225 Table 35.6 Statistic-Option EXACT Statement Statistic-Options Exact Statistics Computed AGREE McNemar s test (for 2 2 tables), simple kappa coefficient test, weighted kappa coefficient test BINOMIAL Binomial proportion tests for one-way tables CHISQ Chi-square goodness-of-fit test for one-way tables; Pearson chi-square, likelihood-ratio chi-square, and Mantel-Haenszel chi-square tests for two-way tables COMOR Confidence limits for the common odds ratio and common odds ratio test (for h 2 2 tables) EQOR ZELEN Zelen s test for equal odds ratios (for h 2 2 tables) FISHER Fisher s exact test JT Jonckheere-Terpstra test KAPPA Test for the simple kappa coefficient KENTB Test for Kendall s tau-b LRCHI Likelihood-ratio chi-square test MCNEM McNemar s test (for 2 2 tables) MEASURES Tests for the Pearson correlation and Spearman correlation, confidence limits for the odds ratio (for 2 2 tables) MHCHI Mantel-Haenszel chi-square test OR Confidence limits for the odds ratio (for 2 2 tables) PCHI Pearson chi-square test PCORR Test for the Pearson correlation coefficient RISKDIFF Confidence limits for the proportion differences (for 2 2 tables) RISKDIFF1 Confidence limits for the column 1 proportion difference RISKDIFF2 Confidence limits for the column 2 proportion difference SCORR Test for the Spearman correlation coefficient SMDCR Test for Somers D.C jr/ SMDRC Test for Somers D.RjC / STUTC Test for Stuart s tau-c TREND Cochran-Armitage test for trend WTKAP Test for the weighted kappa coefficient Using TABLES Statement Options with the EXACT Statement If you use only one TABLES statement, you do not need to specify the same options in both the TABLES and EXACT statements; when you specify a statistic-option in the EXACT statement, PROC FREQ automatically invokes the corresponding TABLES statement option. However, when you use multiple TABLES statements and want exact computations, you must specify options in the TABLES statements to request the desired statistics. PROC FREQ then performs exact computations for all statistics that you also specify in the EXACT statement.

20 2226 Chapter 35: The FREQ Procedure The TABLES statement group option CHISQ includes tests that correspond to the following EXACT statement individual statistic-options: LRCHI, MHCHI, and PCHI. The MEASURES option in the TABLES statement includes statistics that correspond to the following EXACT statement statisticoptions: KENTB, OR, PCORR, SCORR, SMDCR, SMDRC, and STUTC. The AGREE option in the TABLES statement produces analyses that correspond to the KAPPA, MCNEM, and WTKAP statistic-options in the EXACT statement. The CMH option in the TABLES statement produces analyses that correspond to the COMOR and EQOR (ZELEN) statistic-options in the EXACT statement. Computation-Options The computation-options specify options for computation of exact statistics. You can specify the following computation-options in the EXACT statement after a slash (/). ALPHA= specifies the level of the confidence limits for Monte Carlo p-value estimates. The value of must be between 0 and 1, and the default is A confidence level of produces /% confidence limits. The default of ALPHA=.01 produces 99% confidence limits for the Monte Carlo estimates. The ALPHA= option invokes the MC option. MAXTIME=value specifies the maximum clock time (in seconds) that PROC FREQ can use to compute an exact p-value. If the procedure does not complete the computation within the specified time, the computation terminates. The value of MAXTIME= must be a positive number. The MAXTIME= option is valid for Monte Carlo estimation of exact p-values, as well as for direct exact p-value computation. See the section Computational Resources on page 2317 for more information. MC requests Monte Carlo estimation of exact p-values instead of direct exact p-value computation. Monte Carlo estimation can be useful for large problems that require a considerable amount of time and memory for exact computations but for which asymptotic approximations might not be sufficient. See the section Monte Carlo Estimation on page 2317 for more information. The MC option is available for all EXACT statistic-options except the BINOMIAL option and the following options that apply only to 2 2 or h 2 2 tables: COMOR, EQOR, MCNEM, RISKDIFF, and OR. PROC FREQ computes only exact tests or confidence limits for these statistics. The ALPHA=, N=, and SEED= options also invoke the MC option. N=n specifies the number of samples for Monte Carlo estimation. The value of n must be a positive integer, and the default is 10,000. Larger values of n produce more precise estimates of exact p-values. Because larger values of n generate more samples, the computation time increases. The N= option invokes the MC option.

21 OUTPUT Statement 2227 POINT requests exact point probabilities for the test statistics. The POINT option is available for all the EXACT statement statistic-options except the OR and RISKDIFF options, which provide exact confidence limits. The POINT option is not available with the MC option. SEED=number specifies the initial seed for random number generation for Monte Carlo estimation. The value of the SEED= option must be an integer. If you do not specify the SEED= option or if the SEED= value is negative or zero, PROC FREQ uses the time of day from the computer s clock to obtain the initial seed. The SEED= option invokes the MC option. OUTPUT Statement OUTPUT < OUT= SAS-data-set > options ; The OUTPUT statement creates a SAS data set that contains statistics computed by PROC FREQ. You specify which statistics to store in the output data set with the OUTPUT statement options. The output data set contains one observation for each two-way table or stratum, and one observation for summary statistics across all strata. For more information about the contents of the output data set, see the section Contents of the OUTPUT Statement Output Data Set on page Only one OUTPUT statement is allowed for each execution of PROC FREQ. You must specify a TABLES statement with the OUTPUT statement. If you use multiple TABLES statements, the contents of the OUTPUT data set correspond to the last TABLES statement. If you use multiple table requests in a TABLES statement, the contents of the OUTPUT data set correspond to the last table request. Note that you can use the Output Delivery System (ODS) to create a SAS data set from any piece of PROC FREQ output. For more information, see the section ODS Table Names on page Also note that the output data set created by the OUTPUT statement is not the same as the output data set created by the OUT= option in the TABLES statement. The OUTPUT statement creates a data set that contains statistics (such as the Pearson chi-square and its p-value), and the OUT= option in the TABLES statement creates a data set that contains frequency table counts and percentages. See the section Output Data Sets on page 2320 for more information. You can specify the following options in an OUTPUT statement: OUT=SAS-data-set names the output data set. If you omit the OUT= option, the data set is named DATAn, where n is the smallest integer that makes the name unique. options specify the statistics you want in the output data set. Table 35.7 lists the available options, together with the TABLES statement options needed to produce the statistics. You can output groups of statistics by using group options identical to those available in the TABLES statement,

22 2228 Chapter 35: The FREQ Procedure Table 35.7 which include the AGREE, ALL, CHISQ, CMH, and MEASURES options. Or you can request statistics individually. When you specify an option in the OUTPUT statement, the output data set contains all statistics from that analysis the estimate or test statistic plus any associated standard error, confidence limits, p-values, and degrees of freedom. See the section Contents of the OUTPUT Statement Output Data Set on page 2321 for details. If you want to store a statistic in the output data set, you must also request computation of that statistic with the appropriate TABLES or EXACT statement option. For example, you cannot specify the option PCHI (Pearson chi-square) in the OUTPUT statement without also specifying a TABLES or EXACT statement option to compute the Pearson chi-square test. The TABLES statement option ALL or CHISQ requests the Pearson chi-square test. If you have only one TABLES statement, the EXACT statement option CHISQ or PCHI also requests the Pearson chi-square test. Table 35.7 lists the TABLES statement options required to produce the OUTPUT data set statistics. Note that the ALL option in the TABLES statement invokes the CHISQ, MEASURES, and CMH options. OUTPUT Statement Options Option Output Data Set Statistics Required TABLES Statement Option AGREE McNemar s test, Bowker s test, simple and AGREE weighted kappas; for multiple strata, overall simple and weighted kappas, tests for equal kappas, and Cochran s Q AJCHI Continuity-adjusted chi-square (2 2 tables) CHISQ ALL CHISQ, MEASURES, and CMH statistics; N ALL BDCHI Breslow-Day test (h 2 2 tables) CMH, CMH1, or CMH2 BINOMIAL Binomial statistics for one-way tables BINOMIAL CHISQ For one-way tables, goodness-of-fit test; CHISQ for two-way tables, Pearson, likelihood-ratio, continuity-adjusted, and Mantel-Haenszel chi-squares, Fisher s exact test (2 2 tables), phi and contingency coefficients, Cramer s V CMH Cochran-Mantel-Haenszel (CMH) correlation, CMH row mean scores (ANOVA), and general association statistics; for 2 2 tables, logit and Mantel-Haenszel adjusted odds ratios and relative risks, Breslow-Day test CMH1 CMH output, except row mean scores (ANOVA) CMH or CMH1 and general association statistics CMH2 CMH output, except general association statistic CMH or CMH2 CMHCOR CMH correlation statistic CMH, CMH1, or CMH2 CMHGA CMH general association statistic CMH CMHRMS CMH row mean scores (ANOVA) statistic CMH or CMH2 COCHQ Cochran s Q (h 2 2 tables) AGREE CONTGY Contingency coefficient CHISQ CRAMV Cramer s V CHISQ EQKAP Test for equal simple kappas AGREE EQOR ZELEN Zelen s test for equal odds ratios (h 2 2 tables) CMH and EXACT EQOR

23 OUTPUT Statement 2229 Table 35.7 continued Option Output Data Set Statistics Required TABLES Statement Option EQWKP Test for equal weighted kappas AGREE FISHER Fisher s exact test CHISQ or FISHER 1 GAILSIMON Gail-Simon test GAILSIMON GAMMA Gamma MEASURES JT Jonckheere-Terpstra test JT KAPPA Simple kappa coefficient AGREE KENTB Kendall s tau-b MEASURES LAMCR Lambda asymmetric.c jr/ MEASURES LAMDAS Lambda symmetric MEASURES LAMRC Lambda asymmetric.rjc / MEASURES LGOR Adjusted logit odds ratio (h 2 2 tables) CMH, CMH1, or CMH2 LGRRC1 Adjusted column 1 logit relative risk CMH, CMH1, or CMH2 LGRRC2 Adjusted column 2 logit relative risk CMH, CMH1, or CMH2 LRCHI Likelihood-ratio chi-square CHISQ MCNEM McNemar s test (2 2 tables) AGREE Gamma, Kendall s tau-b, Stuart s tau-c, MEASURES MEASURES Somers D.C jr/ and D.RjC /, Pearson and Spearman correlations, lambda asymmetric.c jr/ and.rjc /, lambda symmetric, uncertainty coefficients.c jr/ and.rjc /, symmetric uncertainty coefficient; odds ratio and relative risks (2 2 tables) MHCHI Mantel-Haenszel chi-square CHISQ MHOR COMOR Adjusted Mantel-Haenszel odds ratio CMH, CMH1, or CMH2 (h 2 2 tables) MHRRC1 Adjusted column 1 Mantel-Haenszel relative risk CMH, CMH1, or CMH2 MHRRC2 Adjusted column 2 Mantel-Haenszel relative risk CMH, CMH1, or CMH2 N Number of nonmissing observations NMISS Number of missing observations OR Odds ratio (2 2 tables) MEASURES or RELRISK PCHI Chi-square goodness-of-fit test for one-way tables, CHISQ Pearson chi-square for two-way tables PCORR Pearson correlation coefficient MEASURES PHI Phi coefficient CHISQ PLCORR Polychoric correlation coefficient PLCORR RDIF1 Column 1 risk difference (row 1 - row 2) RISKDIFF RDIF2 Column 2 risk difference (row 1 - row 2) RISKDIFF RELRISK Odds ratio and relative risks (2 2 tables) MEASURES or RELRISK RISKDIFF Risks and risk differences (2 2 tables) RISKDIFF RISKDIFF1 Column 1 risks and risk difference RISKDIFF 1 CHISQ computes Fisher s exact test for 2 2 tables. Use the FISHER option to compute Fisher s exact test for general rxc tables.

24 2230 Chapter 35: The FREQ Procedure Table 35.7 continued Option Output Data Set Statistics Required TABLES Statement Option RISKDIFF2 Column 2 risks and risk difference RISKDIFF RRC1 Column 1 relative risk MEASURES or RELRISK RRC2 Column 2 relative risk MEASURES or RELRISK RSK1 Column 1 risk, overall RISKDIFF RSK11 Column 1 risk, for row 1 RISKDIFF RSK12 Column 2 risk, for row 1 RISKDIFF RSK2 Column 2 risk, overall RISKDIFF RSK21 Column 1 risk, for row 2 RISKDIFF RSK22 Column 2 risk, for row 2 RISKDIFF SCORR Spearman correlation coefficient MEASURES SMDCR Somers D.C jr/ MEASURES SMDRC Somers D.RjC / MEASURES STUTC Stuart s tau-c MEASURES TREND Cochran-Armitage test for trend TREND TSYMM Bowker s test of symmetry AGREE U Symmetric uncertainty coefficient MEASURES UCR Uncertainty coefficient.c jr/ MEASURES URC Uncertainty coefficient.rjc / MEASURES WTKAP Weighted kappa coefficient AGREE TABLES Statement TABLES requests < / options > ; The TABLES statement requests one-way to n-way frequency and crosstabulation tables and statistics for those tables. If you omit the TABLES statement, PROC FREQ generates one-way frequency tables for all data set variables that are not listed in the other statements. The following argument is required in the TABLES statement. requests specify the frequency and crosstabulation tables to produce. A request is composed of one variable name or several variable names separated by asterisks. To request a one-way frequency table, use a single variable. To request a two-way crosstabulation table, use an asterisk between two variables. To request a multiway table (an n-way table, where n>2), separate the desired variables with asterisks. The unique values of these variables form the rows, columns, and strata of the table. You can include up to 50 variables in a single multiway table request. For two-way to multiway tables, the values of the last variable form the crosstabulation table columns, while the values of the next-to-last variable form the rows. Each level (or combination of levels) of the other variables forms one stratum. PROC FREQ produces a separate crosstabulation table for each stratum. For example, a specification of A*B*C*D in

25 TABLES Statement 2231 a TABLES statement produces k tables, where k is the number of different combinations of values for A and B. Each table lists the values for C down the side and the values for D across the top. You can use multiple TABLES statements in the PROC FREQ step. PROC FREQ builds all the table requests in one pass of the data, so that there is essentially no loss of efficiency. You can also specify any number of table requests in a single TABLES statement. To specify multiple table requests quickly, use a grouping syntax by placing parentheses around several variables and joining other variables or variable combinations. For example, the statements shown in Table 35.8 illustrate grouping syntax. Table 35.8 Grouping Syntax Request Equivalent to tables A*(B C); tables A*B A*C; tables (A B)*(C D); tables A*C B*C A*D B*D; tables (A B C)*D; tables A*D B*D C*D; tables A C; tables A B C; tables (A C)*D; tables A*D B*D C*D; The TABLES statement variables are one or more variables from the DATA= input data set. These variables can be either character or numeric, but the procedure treats them as categorical variables. PROC FREQ uses the formatted values of the TABLES variable to determine the categorical variable levels. So if you assign a format to a variable with a FORMAT statement, PROC FREQ formats the values before dividing observations into the levels of a frequency or crosstabulation table. See the discussion of the FORMAT procedure in the Base SAS Procedures Guide and the discussions of the FORMAT statement and SAS formats in SAS Language Reference: Dictionary. If you use PROC FORMAT to create a user-written format that combines missing and nonmissing values into one category, PROC FREQ treats the entire category of formatted values as missing. See the discussion in the section Grouping with Formats on page 2260 for more information. The frequency or crosstabulation table lists the values of both character and numeric variables in ascending order based on internal (unformatted) variable values unless you change the order with the ORDER= option. To list the values in ascending order by formatted value, use ORDER=FORMATTED in the PROC FREQ statement. Without Options If you request a one-way frequency table for a variable without specifying options, PROC FREQ produces frequencies, cumulative frequencies, percentages of the total frequency, and cumulative percentages for each value of the variable. If you request a two-way or an n-way crosstabulation table without specifying options, PROC FREQ produces crosstabulation tables that include cell frequencies, cell percentages of the total frequency, cell percentages of row frequencies, and cell percentages of column frequencies. The procedure excludes observations with missing values from the table but displays the total frequency of missing observations below each table.

26 2232 Chapter 35: The FREQ Procedure Options Table 35.9 lists the options available in the TABLES statement. Descriptions follow in alphabetical order. Table 35.9 Option TABLES Statement Options Description Control Statistical Analysis AGREE Requests tests and measures of classification agreement ALL Requests tests and measures of association produced by CHISQ, MEASURES, and CMH ALPHA= Sets the confidence level for confidence limits BDT Requests Tarone s adjustment for the Breslow-Day test BINOMIAL Requests binomial proportion, confidence limits, and tests for one-way tables BINOMIALC Requests BINOMIAL statistics with a continuity correction CHISQ Requests chi-square tests and measures based on chi-square CL Requests confidence limits for the MEASURES statistics CMH Requests all Cochran-Mantel-Haenszel statistics CMH1 Requests CMH correlation statistic, adjusted odds ratios, and adjusted relative risks CMH2 Requests CMH correlation and row mean scores (ANOVA) statistics, adjusted odds ratios, and adjusted relative risks CONVERGE= Specifies convergence criterion for polychoric correlation FISHER Requests Fisher s exact test for tables larger than 2 2 GAILSIMON Requests Gail-Simon test for qualitative interactions JT Requests Jonckheere-Terpstra test MAXITER= Specifies maximum number of iterations for polychoric correlation MEASURES Requests measures of association MISSING Treats missing values as nonmissing PLCORR Requests polychoric correlation RELRISK Requests relative risk measures for 2 2 tables RISKDIFF Requests risks and risk differences for 2 2 tables RISKDIFFC Requests RISKDIFF statistics with a continuity correction SCORES= Specifies the type of row and column scores TESTF= Specifies expected frequencies for one-way chi-square test TESTP= Specifies expected proportions for one-way chi-square test TREND Requests Cochran-Armitage test for trend Control Additional Table Information CELLCHI2 Displays cell contributions to the Pearson chi-square statistic CUMCOL Displays cumulative column percentages DEVIATION Displays deviations of cell frequencies from expected values EXPECTED Displays expected cell frequencies MISSPRINT Displays missing value frequencies SPARSE Includes all possible combinations of variable levels in LIST and OUT=

27 TABLES Statement 2233 Table 35.9 Option continued Description TOTPCT Displays percentages of total frequency for n-way tables (n > 2) CONTENTS= CROSSLIST FORMAT= LIST NOCOL NOCUM NOFREQ NOPERCENT NOPRINT NOROW NOSPARSE NOWARN PRINTKWT SCOROUT Specifies the contents label for crosstabulation tables Displays crosstabulation tables in ODS column format Formats the frequencies in crosstabulation tables Displays two-way to n-way tables in list format Suppresses display of column percentages Suppresses display of cumulative frequencies and percentages Suppresses display of frequencies Suppresses display of percentages Suppresses display of crosstabulation tables but displays statistics Suppresses display of row percentages Suppresses zero frequency levels in CROSSLIST, LIST and OUT= Suppresses log warning message for the chi-square test Displays kappa coefficient weights Displays row and column scores Produce Statistical Graphics PLOTS= Requests plots from ODS Graphics Create an Output Data Set OUT= Names an output data set to contain frequency counts OUTCUM Includes cumulative frequencies and percentages in the output data set for one-way tables OUTEXPECT Includes expected frequencies in the output data set OUTPCT Includes row, column, and two-way table percentages in the output data set You can specify the following options in a TABLES statement. AGREE < (WT=FC) > requests tests and measures of classification agreement for square tables. The AGREE option provides McNemar s test for 2 2 tables and Bowker s test of symmetry for square tables with more than two response categories. The AGREE option also produces the simple kappa coefficient, the weighted kappa coefficient, their asymptotic standard errors, and their confidence limits. When there are multiple strata, the AGREE option provides overall simple and weighted kappas as well as tests for equal kappas among strata. When there are multiple strata and two response categories, PROC FREQ computes Cochran s Q test. See the section Tests and Measures of Agreement on page 2299 for details about these statistics. If you specify the WT=FC option in parentheses following the AGREE option, PROC FREQ uses Fleiss-Cohen weights to compute the weighted kappa coefficient. By default, PROC FREQ uses Cicchetti-Allison weights. See the section Weighted Kappa Coefficient on page 2301 for details. You can specify the PRINTKWT option to display the kappa coefficient weights.

28 2234 Chapter 35: The FREQ Procedure AGREE statistics are computed only for square tables, where the number of rows equals the number of columns. If your table is not square due to observations with zero weights, you can specify the ZEROS option in the WEIGHT statement to include these observations. For more details, see the section Tables with Zero Rows and Columns on page You can use the TEST statement to request asymptotic tests for the simple and weighted kappa coefficients. You can request exact p-values for the simple and weighted kappa coefficient tests, as well as for McNemar s test, by specifying the corresponding options in the EXACT statement. See the section Exact Statistics on page 2314 for more information. ALL requests all of the tests and measures that are computed by the CHISQ, MEASURES, and CMH options. The number of CMH statistics computed can be controlled by the CMH1 and CMH2 options. ALPHA= specifies the level of confidence limits. The value of must be between 0 and 1, and the default is A confidence level of produces /% confidence limits. The default of ALPHA=0.05 produces 95% confidence limits. ALPHA= applies to confidence limits requested by TABLES statement options. There is a separate ALPHA= option in the EXACT statement that sets the level of confidence limits for Monte Carlo estimates of exact p-values, which are requested in the EXACT statement. BDT requests Tarone s adjustment in the Breslow-Day test for homogeneity of odds ratios. (You must specify the CMH option to compute the Breslow-Day test.) See the section Breslow-Day Test for Homogeneity of the Odds Ratios on page 2310 for more information. BINOMIAL < (binomial-options) > requests the binomial proportion for one-way tables. When you specify the BINOMIAL option, PROC FREQ also provides the asymptotic standard error, asymptotic (Wald) and exact (Clopper-Pearson) confidence limits, and the asymptotic equality test for the binomial proportion by default. You can specify binomial-options inside the parentheses following the BINOMIAL option. The LEVEL= binomial-option identifies the variable level for which to compute the proportion. If you do not specify LEVEL=, PROC FREQ computes the proportion for the first level that appears in the output. The P= binomial-option specifies the null proportion for the binomial tests. If you do not specify P=, PROC FREQ uses P=0.5 by default. You can also specify binomial-options to request confidence limits and tests of noninferiority, superiority, and equivalence for the binomial proportion. Table summarizes the binomialoptions. Available confidence limits for the binomial proportion include Agresti-Coull, exact (Clopper- Pearson), Jeffreys, Wald, and Wilson (score) confidence limits. You can specify more than one type of binomial confidence limits in the same analysis. If you do not specify any confidence limit requests with binomial-options, PROC FREQ computes Wald asymptotic confidence limits and exact (Clopper-Pearson) confidence limits by default. The ALPHA= option determines the confidence level, and the default of ALPHA=0.05 produces 95% confidence limits for the binomial proportion.

29 TABLES Statement 2235 To request exact tests for the binomial proportion, specify the BINOMIAL option in the EXACT statement. PROC FREQ then computes exact p-values for all binomial tests that you request with binomial-options, which can include tests of noninferiority, superiority, and equivalence, as well as the test of equality. See the section Binomial Proportion on page 2280 for details. Table BINOMIAL Options Task Specify the variable level Specify the null proportion P= Request a continuity correction Request confidence limits Request tests Specify the test margin Specify the test variance Binomial-Option LEVEL= CORRECT AGRESTICOULL AC ALL EXACT CLOPPERPEARSON JEFFREYS J WILSON W WALD EQUIV EQUIVALENCE NONINF NONINFERIORITY SUP SUPERIORITY MARGIN= VAR=SAMPLE NULL You can specify the following binomial-options inside parentheses following the BINOMIAL option. AGRESTICOULL AC requests Agresti-Coull confidence limits for the binomial proportion. See the section Agresti-Coull Confidence Limits on page 2280 for details. ALL requests all available types of confidence limits for the binomial proportion. These include the following: Agresti-Coull, exact (Clopper-Pearson), Jeffreys, Wald, and Wilson (score) confidence limits. CORRECT includes a continuity correction in the asymptotic Wald confidence limits and tests. The CORRECT binomial-option has the same effect as the BINOMIALC option. EQUIV EQUIVALENCE requests a test of equivalence for the binomial proportion. See the section Equivalence Test on page 2285 for details. You can specify the equivalence test margins, the null proportion, and the variance type with the MARGIN=, P=, and VAR= binomial-options, respectively.

30 2236 Chapter 35: The FREQ Procedure EXACT CLOPPERPEARSON requests exact (Clopper-Pearson) confidence limits for the binomial proportion. See the section Exact (Clopper-Pearson) Confidence Limits on page 2281 for details. If you do not request any binomial confidence limits by specifying binomial-options, PROC FREQ produces Wald and exact (Clopper-Pearson) confidence limits by default. JEFFREYS J requests Jeffreys confidence limits for the binomial proportion. See the section Jeffreys Confidence Limits on page 2281 for details. LEVEL=level-number level-value specifies the variable level for the binomial proportion. By default, PROC FREQ computes the proportion of observations for the first variable level that appears in the output. To request a different level, use LEVEL=level-number or LEVEL= level-value, where level-number is the variable level s number or order in the output, and level-value is the formatted value of the variable level. The value of level-number must be a positive integer. You must enclose level-value in single quotes. MARGIN=value (lower,upper) specifies the margin for the noninferiority, superiority, and equivalence tests, which you request with the NONINF, SUP, and EQUIV binomial-options, respectively. If you do not specify MARGIN=, PROC FREQ uses a margin of 0.2 by default. For noninferiority and superiority tests, specify a single value for the MARGIN= option. The MARGIN= value must be a positive number. You can specify value as a number between 0 and 1. Or you can specify value in percentage form as a number between 1 and 100, and PROC FREQ converts that number to a proportion. The procedure treats the value 1 as 1%. For noninferiority and superiority tests, the test limits must be between 0 and 1. The limits are determined by the null proportion value (which you can specify with the P= binomial-option) and by the margin value. The noninferiority limit equals the null proportion minus the margin. By default, the null proportion equals 0.5 and the margin equals 0.2, which gives a noninferiority limit of 0.3. The superiority limit equals the null proportion plus the margin, which is 0.7 by default. For an equivalence test, you can specify a single MARGIN= value, or you can specify both lower and upper values. If you specify a single MARGIN= value, it must be a positive number, as described previously. If you specify a single MARGIN= value for an equivalence test, PROC FREQ uses value as the lower margin and value as the upper margin for the test. If you specify both lower and upper values for an equivalence test, you can specify them in proportion form as numbers between 1 or 1. Or you can specify them in percentage form as numbers between 100 and 100, and PROC FREQ converts the numbers to proportions. The value of lower must be less than the value of upper. The equivalence limits must be between 0 and 1. The equivalence limits are determined by the null proportion value (which you can specify with the P= binomial-option) and by the margin values. The lower equivalence limit equals the null proportion plus the

31 TABLES Statement 2237 lower margin. By default, the null proportion equals 0.5 and the lower margin equals 0.2, which gives a lower equivalence limit of 0.3. The upper equivalence limit equals the null proportion plus the upper margin, which is 0.7 by default. See the sections Noninferiority Test on page 2283 and Equivalence Test on page 2285 for details. NONINF NONINFERIORITY requests a test of noninferiority for the binomial proportion. See the section Noninferiority Test on page 2283 for details. You can specify the noninferiority test margin, the null proportion, and the variance type with the MARGIN=, P=, and VAR= binomial-options, respectively. P=value specifies the null hypothesis proportion for the binomial tests. If you omit the P= option, PROC FREQ uses 0.5 as the null proportion. The null proportion value must be a positive number. You can specify value as a number between 0 and 1. Or you can specify value in percentage form as a number between 1 and 100, and PROC FREQ converts that number to a proportion. The procedure treats the value 1 as 1%. SUP SUPERIORITY requests a test of superiority for the binomial proportion. See the section Superiority Test on page 2284 for details. You can specify the superiority test margin, the null proportion, and the variance type with the MARGIN=, P=, and VAR= binomial-options, respectively. VAR=SAMPLE NULL specifies the type of variance estimate to use for the tests of noninferiority, superiority, and equivalence. The default is VAR=SAMPLE, which estimates the variance from the sample proportion. VAR=NULL uses the null proportion to compute the variance. See the sections Noninferiority Test on page 2283 and Equivalence Test on page 2285 for details. WALD requests Wald confidence limits for the binomial proportion. See the section Wald Confidence Limits on page 2280 for details. If you do not request any binomial confidence limits by specifying binomial-options, PROC FREQ produces Wald and exact (Clopper-Pearson) confidence limits by default. WILSON W SCORE requests Wilson confidence limits for the binomial proportion. These are also known as score confidence limits. See the section Wilson (Score) Confidence Limits on page 2281 for details. BINOMIALC < (binomial-options) > requests the BINOMIAL option statistics for one-way tables, and includes a continuity correction in the asymptotic Wald confidence limits and tests. The BINOMIAL option statistics include the binomial proportion, its asymptotic standard error, asymptotic (Wald) and exact

32 2238 Chapter 35: The FREQ Procedure (Clopper-Pearson) confidence limits, and the asymptotic equality test for the binomial proportion by default. The binomial-options available with the BINOMIALC option are the same as those available with BINOMIAL. See the description of the BINOMIAL option for details. CELLCHI2 displays each crosstabulation table cell s contribution to the total Pearson chi-square statistic. The cell contribution is computed as.frequency expected/ 2 expected where frequency is the table cell frequency or count and expected is the expected cell frequency, which is computed under the null hypothesis that the row and column variables are independent. See the section Pearson Chi-Square Test for Two-Way Tables on page 2267 for details. The CELLCHI2 option has no effect for one-way tables or for tables that are displayed with the LIST option. CHISQ < ( option ) > requests chi-square tests of homogeneity or independence and measures of association based on the chi-square statistic. The tests include the Pearson chi-square, likelihood-ratio chi-square, and Mantel-Haenszel chi-square. The measures include the phi coefficient, the contingency coefficient, and Cramer s V. For 2 2 tables, the CHISQ option also provides Fisher s exact test and the continuity-adjusted chi-square. See the section Chi-Square Tests and Statistics on page 2265 for details. For one-way tables, the CHISQ option provides a chi-square goodness-of-fit test for equal proportions. If you specify the null hypothesis proportions with the TESTP= option, PROC FREQ computes a chi-square goodness-of-fit test for the specified proportions. If you specify null hypothesis frequencies with the TESTF= option, PROC FREQ computes a chi-square goodness-of-fit test for the specified frequencies. See the section Chi-Square Test for One-Way Tables on page 2266 for more information. To request Fisher s exact test for tables larger than 2 2, use the FISHER option in the EXACT statement. Exact tests are also available for other CHISQ statistics, including the Pearson, likelihood-ratio, and Mantel-Haenszel chi-square, and the chi-square goodness-of-fit test for one-way tables. You can use the EXACT statement to request these tests. See the section Exact Statistics on page 2314 for details. You can specify the following option in parentheses following the CHISQ option: WARN= value WARN= ( values ) controls the warning message about the validity of the asymptotic Pearson chi-square test. By default, PROC FREQ displays a warning when more than 20% of the table cells have expected frequencies that are less than 5. If you specify the NOPRINT option in the PROC FREQ statement, this warning is included in the log; otherwise, the warning is displayed as a footnote in the chi-square table. You can use the WARN= option to suppress the warning and to include a warning indicator in the output data set.

33 TABLES Statement 2239 The WARN= option can take one or more of the following values. If you specify more than one value, enclose the values in parentheses following WARN=. For example, warn = (output noprint). Value of WARN= OUTPUT NOLOG NOPRINT NONE Description Adds a warning indicator variable to the output data set Suppresses the chi-square warning message in the log Suppresses the chi-square warning message in the display Suppresses the chi-square warning message entirely If you specify the WARN=OUTPUT option, the chi-square ODS output data set contains a variable named Warning that equals 1 for the Pearson chi-square when more than 20% of the table cells have expected frequencies that are less than 5 and equals 0 otherwise. If you specify WARN=OUTPUT and also specify the CHISQ option in the OUTPUT statement, then the statistics output data set contains a variable named WARN_PCHI that indicates the warning. The WARN=NOLOG option has the same effect as the NOWARN option in the TABLES statement. CL requests confidence limits for the MEASURES statistics. If you omit the MEASURES option, the CL option invokes MEASURES. You can set the level of the confidence limits by using the ALPHA= option. The default of ALPHA=0.5 produces 95% confidence limits. See the sections Measures of Association on page 2270 and Confidence Limits on page 2271 for more information. CMH < ( cmh-options ) > requests Cochran-Mantel-Haenszel statistics, which test for association between the row and column variables after adjusting for the remaining variables in a multiway table. The Cochran- Mantel-Haenszel statistics include the nonzero correlation statistic, the row mean scores (ANOVA) statistic, and the general association statistic. In addition, for 2 2 tables, the CMH option provides the adjusted Mantel-Haenszel and logit estimates of the odds ratio and relative risks, together with their confidence limits. For stratified 2 2 tables, the CMH option provides the Breslow-Day test for homogeneity of odds ratios. (To request Tarone s adjustment for the Breslow-Day test, specify the BDT cmh-option.) See the section Cochran-Mantel-Haenszel Statistics on page 2304 for details. You can use the CMH1 or CMH2 option to control the number of CMH statistics that PROC FREQ computes. For stratified 2 2 tables, you can request Zelen s exact test for equal odds ratios by specifying the EQOR option in the EXACT statement. See the section Zelen s Exact Test for Equal Odds Ratios on page 2311 for details. You can request exact confidence limits for the common odds ratio by specifying the COMOR option in the EXACT statement. This option also provides a common odds ratio test. See the section Exact Confidence Limits for the Common Odds Ratio on page 2311 for details.

34 2240 Chapter 35: The FREQ Procedure You can specify the following cmh-options in parentheses following the CMH option. These cmh-options, which apply to stratified 2 2 tables, are also available with the CMH1 or CMH2 option. BDT requests Tarone s adjustment in the Breslow-Day test for homogeneity of odds ratios. See the section Breslow-Day Test for Homogeneity of the Odds Ratios on page 2310 for details. The BDT cmh-option has the same effect as the BDT option in the TABLES statement. GAILSIMON GS < COLUMN=1 2 > requests the Gail-Simon test for qualitative interaction, which applies to stratified 2 2 tables. See the section Gail-Simon Test for Qualitative Interactions on page 2313 for details. The COLUMN= option specifies the column of the risk differences to use in computing the Gail-Simon test. By default, PROC FREQ uses column 1 risk differences. If you specify COLUMN=2, PROC FREQ uses column 2 risk differences. The GAILSIMON cmh-option has the same effect as the GAILSIMON option in the TABLES statement. MANTELFLEISS MF requests the Mantel-Fleiss criterion for the Mantel-Haenszel statistic for stratified 2 2 tables. See the section Mantel-Fleiss Criterion on page 2307 for details. CMH1 < ( cmh-options ) > requests the Cochran-Mantel-Haenszel correlation statistic. This option does not provide the CMH row mean scores (ANOVA) statistic or the general association statistic, which are provided by the CMH option. For tables larger than 2 2, the CMH1 option requires less memory than the CMH option, which can require an enormous amount of memory for large tables. For 2 2 tables, the CMH1 option also provides the adjusted Mantel-Haenszel and logit estimates of the odds ratio and relative risks, together with their confidence limits. For stratified 2 2 tables, the CMH1 option provides the Breslow-Day test for homogeneity of odds ratios. The cmh-options available with the CMH1 option are the same as those available with the CMH option. See the description of the CMH option for details. CMH2 < ( cmh-options ) > requests the Cochran-Mantel-Haenszel correlation statistic and the row mean scores (ANOVA) statistic. This option does not provide the CMH general association statistic, which is provided by the CMH option. For tables larger than 2 2, the CMH2 option requires less memory than the CMH option, which can require an enormous amount of memory for large tables. For 2 2 tables, the CMH1 option also provides the adjusted Mantel-Haenszel and logit estimates of the odds ratio and relative risks, together with their confidence limits. For stratified 2 2 tables, the CMH1 option provides the Breslow-Day test for homogeneity of odds ratios.

35 TABLES Statement 2241 The cmh-options available with the CMH2 option are the same as those available with the CMH option. See the description of the CMH option for details. CONTENTS= string specifies the label to use for crosstabulation tables in the contents file, the Results window, and the trace record. For information about output presentation, see the SAS Output Delivery System: User s Guide. If you omit the CONTENTS= option, the contents label for crosstabulation tables is Cross- Tabular Freq Table by default. Note that contents labels for all crosstabulation tables that are produced by a single TABLES statement use the same text. To specify different contents labels for different crosstabulation tables, request the tables in separate TABLES statements and use the CONTENTS= option in each TABLES statement. To remove the crosstabulation table entry from the contents file, you can specify a null label with CONTENTS=. The CONTENTS= option affects only contents labels for crosstabulation tables. It does not affect contents labels for other PROC FREQ tables. To specify the contents label for any PROC FREQ table, you can use PROC TEMPLATE to create a customized table definition. The CONTENTS_LABEL attribute in the DEFINE TABLE statement of PROC TEMPLATE specifies the contents label for the table. See the chapter The TEMPLATE Procedure in the SAS Output Delivery System: User s Guide for more information. CONVERGE=value specifies the convergence criterion for computing the polychoric correlation, which you request with the PLCORR option. The CONVERGE= value must be a positive number. By default CONVERGE= Iterative computation of the polychoric correlation stops when the convergence measure falls below the value of CONVERGE= or when the number of iterations exceeds the value specified in the MAXITER= option, whichever happens first. See the section Polychoric Correlation on page 2277 for details. CROSSLIST displays crosstabulation tables in ODS column format instead of the default crosstabulation cell format. In a CROSSLIST table display, the rows correspond to the crosstabulation table cells, and the columns correspond to descriptive statistics such as Frequency and Percent. The CROSSLIST table displays the same information as the default crosstabulation table, but uses an ODS column format instead of the table cell format. See the section Multiway Tables on page 2324 for details about the contents of the CROSSLIST table. You can control the contents of a CROSSLIST table with the same options available for the default crosstabulation table. These include the NOFREQ, NOPERCENT, NOROW, and NOCOL options. You can request additional information in a CROSSLIST table with the CELLCHI2, DEVIATION, EXPECTED, MISSPRINT, and TOTPCT options. The FORMAT= option and the CUMCOL option have no effect for CROSSLIST tables. You cannot specify both the LIST option and the CROSSLIST option in the same TABLES statement.

36 2242 Chapter 35: The FREQ Procedure You can use the NOSPARSE option to suppress display of variable levels with zero frequency in CROSSLIST tables. By default for CROSSLIST tables, PROC FREQ displays all levels of the column variable within each level of the row variable, including any column variable levels with zero frequency for that row. And for multiway tables displayed with the CROSSLIST option, the procedure displays all levels of the row variable for each stratum of the table by default, including any row variable levels with zero frequency for the stratum. CUMCOL displays the cumulative column percentages in the cells of the crosstabulation table. The CUM- COL option does not apply to crosstabulation tables produced with the LIST or CROSSLIST option. DEVIATION displays the deviation of the frequency from the expected frequency for each cell of the crosstabulation table. See the section Pearson Chi-Square Test for Two-Way Tables on page 2267 for details. The DEVIATION option does not apply to crosstabulation tables produced with the LIST option. EXPECTED displays the expected cell frequencies under the hypothesis of independence (or homogeneity) for crosstabulation tables. See the section Pearson Chi-Square Test for Two-Way Tables on page 2267 for details. The EXPECTED option does not apply to tables produced with the LIST option. FISHER EXACT requests Fisher s exact test for tables that are larger than 2 2. (For 2 2 tables, the CHISQ option provides Fisher s exact test.) This test is also known as the Freeman-Halton test. See the sections Fisher s Exact Test on page 2268 and Exact Statistics on page 2314 for more information. If you omit the CHISQ option in the TABLES statement, the FISHER option invokes CHISQ. You can also request Fisher s exact test by specifying the FISHER option in the EXACT statement. NOTE: PROC FREQ computes exact tests with fast and efficient algorithms that are superior to direct enumeration. Exact tests are appropriate when a data set is small, sparse, skewed, or heavily tied. For some large problems, computation of exact tests might require a considerable amount of time and memory. Consider using asymptotic tests for such problems. Alternatively, when asymptotic methods might not be sufficient for such large problems, consider using Monte Carlo estimation of exact p-values. See the section Computational Resources on page 2317 for more information. FORMAT=format-name specifies a format for the following crosstabulation table cell values: frequency, expected frequency, and deviation. PROC FREQ also uses the specified format to display the row and column total frequencies and the overall total frequency in crosstabulation tables. You can specify any standard SAS numeric format or a numeric format defined with the FORMAT procedure. The format length must not exceed 24. If you omit the FORMAT= option, by default PROC FREQ uses the BEST6. format to display frequencies less than 1E6, and the BEST7. format otherwise.

37 TABLES Statement 2243 The FORMAT= option applies only to crosstabulation tables displayed in the default format. It does not apply to crosstabulation tables produced with the LIST or CROSSLIST option. To change display formats in any FREQ table, you can use PROC TEMPLATE. See the chapter The TEMPLATE Procedure in the SAS Output Delivery System: User s Guide for more information. GAILSIMON GS < COLUMN = 1 2 > requests the Gail-Simon test for qualitative interaction, which applies to stratified 2 2 tables. See the section Gail-Simon Test for Qualitative Interactions on page 2313 for details. The COLUMN= option specifies the column of the risk differences to use in computing the Gail-Simon test. By default, PROC FREQ uses column 1 risk differences. If you specify COLUMN=2, PROC FREQ uses column 2 risk differences. JT LIST requests the Jonckheere-Terpstra test. See the section Jonckheere-Terpstra Test on page 2297 for details. displays two-way to n-way crosstabulation tables in a list format instead of the default crosstabulation cell format. The LIST option displays the entire multiway table in one table, instead of displaying a separate two-way table for each stratum. See the section Multiway Tables on page 2324 for details. The LIST option is not available when you also specify statistical options. You must use the standard crosstabulation table display or the CROSSLIST display when you request statistical tests or measures. MAXITER=number specifies the maximum number of iterations for computing the polychoric correlation, which you request with the PLCORR option. The value of the MAXITER= option must be a positive integer. By default MAXITER=20. Iterative computation of the polychoric correlation stops when the number of iterations exceeds the MAXITER= value or when the convergence measures falls below the value of the CONVERGE= option, whichever happens first. See the section Polychoric Correlation on page 2277 for details. MEASURES requests several measures of association and their asymptotic standard errors. The MEA- SURES option provides the following statistics: gamma, Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, Somers D.RjC /, the Pearson and Spearman correlation coefficients, lambda (symmetric and asymmetric), and uncertainty coefficients (symmetric and asymmetric). To request confidence limits for these measures of association, you can specify the CL option. For 2 2 tables, the MEASURES option also provides the odds ratio, column 1 relative risk, column 2 relative risk, and the corresponding confidence limits. Alternatively, you can obtain the odds ratio and relative risks, without the other measures of association, by specifying the RELRISK option. See the section Measures of Association on page 2270 for details.

38 2244 Chapter 35: The FREQ Procedure You can use the TEST statement to request asymptotic tests for the following measures of association: gamma, Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, Somers D.RjC /, and the Pearson and Spearman correlation coefficients. You can use the EXACT statement to request exact confidence limits for the odds ratio and exact tests for the following measures of association: Kendall s tau-b, Stuart s tau-c, Somers D.C jr/ and D.RjC /, and the Pearson and Spearman correlation coefficients. See the section Exact Statistics on page 2314 for more information. MISSING treats missing values as a valid nonmissing level for all TABLES variables. The MISSING option displays the missing levels in frequency and crosstabulation tables and includes them in all calculations of percentages, tests, and measures. By default, if you do not specify the MISSING or MISSPRINT option, an observation is excluded from a table if it has a missing value for any of the variables in the TABLES request. When PROC FREQ excludes observations with missing values, it displays the total frequency of missing observations below the table. See the section Missing Values on page 2261 for more information. MISSPRINT displays missing value frequencies in frequency and crosstabulation tables but does not include the missing value frequencies in any computations of percentages, tests, or measures. By default, if you do not specify the MISSING or MISSPRINT option, an observation is excluded from a table if it has a missing value for any of the variables in the TABLES request. When PROC FREQ excludes observations with missing values, it displays the total frequency of missing observations below the table. See the section Missing Values on page 2261 for more information. NOCOL suppresses the display of column percentages in crosstabulation table cells. NOCUM suppresses the display of cumulative frequencies and percentages in one-way frequency tables. The NOCUM option also suppresses the display of cumulative frequencies and percentages in crosstabulation tables in list format, which you request with the LIST option. NOFREQ suppresses the display of cell frequencies in crosstabulation tables. The NOFREQ option also suppresses row total frequencies. This option has no effect for one-way tables or for crosstabulation tables in list format, which you request with the LIST option. NOPERCENT suppresses the display of overall percentages in crosstabulation tables. These percentages include the cell percentages of the total (two-way) table frequency, as well as the row and column percentages of the total table frequency. To suppress the display of cell percentages of row or column totals, use the NOROW or NOCOL option, respectively. For one-way frequency tables and crosstabulation tables in list format, the NOPERCENT option suppresses the display of percentages and cumulative percentages.

39 TABLES Statement 2245 NOPRINT suppresses the display of frequency and crosstabulation tables but displays all requested tests and statistics. To suppress the display of all output, including tests and statistics, use the NOPRINT option in the PROC FREQ statement. NOROW suppresses the display of row percentages in crosstabulation table cells. NOSPARSE suppresses the display of cells with a zero frequency count in LIST output and omits them from the OUT= data set. The NOSPARSE option applies when you specify the ZEROS option in the WEIGHT statement to include observations with zero weights. By default, the ZEROS option invokes the SPARSE option, which displays table cells with a zero frequency count in the LIST output and includes them in the OUT= data set. See the description of the ZEROS option for more information. The NOSPARSE option also suppresses the display of variable levels with zero frequency in CROSSLIST tables. By default for CROSSLIST tables, PROC FREQ displays all levels of the column variable within each level of the row variable, including any column variable levels with zero frequency for that row. For multiway tables displayed with the CROSSLIST option, the procedure displays all levels of the row variable for each stratum of the table by default, including any row variable levels with zero frequency for the stratum. NOWARN suppresses the log warning message about the validity of the asymptotic Pearson chi-square test. By default, PROC FREQ provides a warning about the validity of the asymptotic Pearson chi-square test when more than 20cells have expected frequencies that are less than 5. This warning message appears in the log if you specify the NOPRINT option in the PROC FREQ statement, The NOWARN option is equivalent to the CHISQ(WARN=NOLOG) option. You can also use the CHISQ(WARN=) option to suppress the warning message in the display and to request a warning variable in the chi-square ODS output data set or in the OUTPUT data set. OUT=SAS-data-set names an output data set that contains frequency or crosstabulation table counts and percentages. If more than one table request appears in the TABLES statement, the contents of the OUT= data set correspond to the last table request in the TABLES statement. The OUT= data set variable COUNT contains the frequencies and the variable PERCENT contains the percentages. See the section Output Data Sets on page 2320 for details. You can specify the following options to include additional information in the OUT= data set: OUTCUM, OUTEXPECT, and OUTPCT. OUTCUM includes cumulative frequencies and cumulative percentages in the OUT= data set for oneway tables. The variable CUM_FREQ contains the cumulative frequencies, and the variable CUM_PCT contains the cumulative percentages. See the section Output Data Sets on page 2320 for details. The OUTCUM option has no effect for two-way or multiway tables.

40 2246 Chapter 35: The FREQ Procedure OUTEXPECT includes expected cell frequencies in the OUT= data set for crosstabulation tables. The variable EXPECTED contains the expected cell frequencies. See the section Output Data Sets on page 2320 for details. The EXPECTED option has no effect for one-way tables. OUTPCT includes the following additional variables in the OUT= data set for crosstabulation tables: PCT_COL PCT_ROW PCT_TABL percentage of column frequency percentage of row frequency percentage of stratum (two-way table) frequency, for n-way tables where n > 2 See the section Output Data Sets on page 2320 for details. The OUTPCT option has no effect for one-way tables. PLCORR requests the polychoric correlation coefficient. For 2 2 tables, this statistic is more commonly known as the tetrachoric correlation coefficient, and it is labeled as such in the displayed output. See the section Polychoric Correlation on page 2277 for details. Also see the descriptions of the CONVERGE= and MAXITER= options, which you can specify to control the iterative computation of the polychoric correlation coefficient. If you omit the MEASURES option, the PLCORR option invokes MEASURES. PLOTS < ( global-plot-options ) > < = plot-request < ( plot-options ) > > PLOTS < ( global-plot-options ) > < = ( plot-request < (plot-options) > <... plot-request < (plot-options ) > > ) > controls the plots that are produced through ODS Graphics. Plot-requests identify the plots, and plot-options control the appearance and content of the plots. You can specify plot-options in parentheses following a plot-request. A global-plot-option applies to all plots for which it is available, unless it is altered by a specific plot-option. You can specify global-plot-options in parentheses following the PLOTS option. When you specify only one plot-request, you can omit the parentheses around the request. For example: plots=all plots=freqplot plots=(freqplot oddsratioplot) plots(only)=(cumfreqplot deviationplot) For information about ODS Graphics, see Chapter 21, Statistical Graphics Using ODS. You must enable ODS Graphics before requesting plots, as shown in the following example: ods graphics on; proc freq; tables treatment*response / chisq plots=freqplot; weight wt; run; ods graphics off;

41 TABLES Statement 2247 The PLOTS= option has no effect when you specify the NOPRINT option in the PROC FREQ statement. If you have enabled ODS Graphics but do not specify the PLOTS= option, then PROC FREQ produces all plots that are associated with the analyses that you request in the TABLES statement. Table lists the available plot-requests, together with their plot-options and required TABLES statement options. Table PLOTS= Options Plot-Request Plot-Options Required TABLES Statement Option CUMFREQPLOT ORIENT= One-way table request SCALE= TYPE= DEVIATIONPLOT NOSTAT CHISQ (one-way table) ORIENT= TYPE= FREQPLOT ORIENT= Any table request SCALE= TYPE= FREQPLOT NPANELPOS= Two-way or multiway table TWOWAY= KAPPAPLOT NPANELPOS= AGREE (h r r table) ORDER= RANGE= STATS ODDSRATIOPLOT EXACT MEASURES or RELRISK LOGBASE= (h 2 2 table) NPANELPOS= ORDER= RANGE= STATS RELRISKPLOT LOGBASE= MEASURES or RELRISK NPANELPOS= (h 2 2 table) ORDER= RANGE= STATS RISKDIFFPLOT EXACT RISKDIFF NPANELPOS= (h 2 2 table) ORDER= RANGE= STATS Also requires EXACT statement

42 2248 Chapter 35: The FREQ Procedure Table continued Plot-Request Plot-Options Required TABLES Statement Option WTKAPPAPLOT NPANELPOS= AGREE ORDER= (h r r table, r > 2) RANGE= STATS Global-Plot-Options A global-plot-option applies to all plots for which the option is available, unless it is altered by a specific plot-option. You can specify global-plot-options in parentheses following the PLOTS option. The following specific plot-options are available as global-plot-options: COLUMN=, EXACT, LOGBASE=, NPANELPOS=, ORDER=, ORIENT= RANGE=, SCALE=, STATS, and TYPE=. These plot-options are described in the section Plot-Options. Additionally, you can specify the following global-plot-option in parentheses following the PLOTS option: ONLY suppresses the default plots and requests only the plots that are specified as plot-requests. Plot-Requests The following plot-requests are available with the PLOTS= option: ALL requests all plots that are associated with the specified analyses. This is the default if you do not specify the PLOTS(ONLY) option. CUMFREQPLOT < ( plot-options ) > requests a plot of cumulative frequencies. Cumulative frequency plots are available for one-way frequency tables. The following plot-options are available for CUMFREQPLOT: ORIENT=, SCALE=, and TYPE=. DEVIATIONPLOT < ( plot-options ) > requests a plot of relative deviations from expected frequencies. Deviation plots are available for one-way frequency tables. To produce a deviation plot, you must also specify the CHISQ option in the TABLES statement. The following plot-options are available for DEVIATIONPLOT: NOSTAT, ORIENT= and TYPE=. FREQPLOT < ( plot-options ) > requests a frequency plot. Frequency plots are available for frequency and crosstabulation tables. For multiway tables, PROC FREQ provides a two-way frequency plot for each stratum.

43 TABLES Statement 2249 The following plot-options are available for FREQPLOT for all tables: ORIENT=, SCALE=, and TYPE=. Additionally, the TWOWAY= and NPANELPOS= plot-options are available for two-way and multiway tables. You can use the TWOWAY= plot-option to specify the layout of a two-way frequency plot. The NPANELPOS= plot-option is not available with the TWOWAY=STACKED layout. KAPPAPLOT < ( plot-options ) > requests a plot of kappa statistics and confidence limits. Kappa plots are available for multiway square tables. To produce a kappa plot, you must also specify the AGREE option in the TABLES statement. The following plot-options are available for KAPPAPLOT: NPANELPOS=, ORDER=, RANGE=, and STATS. NONE suppresses all plots. ODDSRATIOPLOT < ( plot-options ) > requests a plot of odds ratios with confidence limits. Odds ratio plots are available for multiway 2 2 tables. To produce an odds ratio plot, you must also specify the MEASURES or RELRISK option in the TABLES statement. The following plot-options are available for ODDSRATIOPLOT: EXACT, LOGBASE=, NPANELPOS=, ORDER=, RANGE=, and STATS. If you request a plot of exact confidence limits by specifying the EXACT plot-option, you must also request computation of exact confidence limits by specifying the EXACT statement. RELRISKPLOT < ( plot-options ) > requests a plot of relative risks with confidence limits. Relative risk plots are available for multiway 2 2 tables. To produce a relative risk plot, you must also specify the MEASURES or RELRISK option in the TABLES statement. The following plotoptions are available for RELRISKPLOT: COLUMN=, LOGBASE=, NPANELPOS=, ORDER=, RANGE=, and STATS. RISKDIFFPLOT < ( plot-options ) > requests a plot of risk (proportion) differences with confidence limits. Risk difference plots are available for multiway 2 2 tables. To produce a risk difference plot, you must also specify the RISKDIFF option in the TABLES statement. The following plotoptions are available for RISKDIFF: COLUMN=, EXACT, NPANELPOS=, ORDER=, RANGE=, and STATS. If you request a plot of exact confidence limits by specifying the EXACT plot-option, you must also request computation of exact confidence limits by specifying the EXACT statement. WTKAPPAPLOT < ( plot-options ) > requests a plot of weighted kappa statistics with confidence limits. Weighted kappa plots are available for multiway square tables. To produce a weighted kappa plot, you must also specify the AGREE option in the TABLES statement. Note that simple kappa and weighted kappa statistics are the same for 2 2 tables; therefore, the procedure does not present weighted kappa statistics for 2 2 tables. The following plot-options are available for WTKAPPAPLOT: NPANELPOS=, ORDER=, RANGE=, and STATS.

44 2250 Chapter 35: The FREQ Procedure Plot-Options You can specify the following plot-options in parentheses after a plot-request: COLUMN=1 2 specifies the risk column for the relative risk plot (RELRISKPLOT) and the risk difference plot (RISKDIFFPLOT). If you specify the COLUMN=1 plot-option, then the plot displays the column 1 risk differences or the column 1 relative risks. Similarly, COLUMN=2 displays the column 2 risk differences or relative risks. The default is COLUMN=1. EXACT requests exact confidence limits instead of asymptotic confidence limits in the odds ratio plot (ODDSRATIOPLOT) and the risk difference plot (RISKDIFFPLOT). If you specify the EXACT plot-option, you must also request computation of the exact confidence limits by specifying the EXACT statement. LOGBASE=2 E 10 applies to the odds ratio plot (ODDSRATIOPLOT) and the relative risk plot (RELRISKPLOT). The LOGBASE= plot-option displays the odds ratio or relative risk axis on the specified log scale. NOSTAT applies to the deviation plot (DEVIATIONPLOT). The NOSTAT plot-option suppresses the chi-square p-value that is displayed by default in the deviation plot. NPANELPOS=n applies to the following plots: FREQPLOT (for two-way and multiway tables), KAPPAPLOT, ODDSRATIOPLOT, RELRISKPLOT, RISKDIFFPLOT, and WTKAPPAPLOT. The NPANELPOS= plot-option divides the plot into multiple panels that display at most jnj statistics per panel. If n is positive, the number of statistics per panel is balanced; but if n is negative, the number of statistics per panel is not balanced. By default, n D 0 and all statistics are displayed in a single plot. For example, suppose you want to display 21 odds ratios. Then NPANELPOS=20 displays two panels, the first with 11 odds ratios and the second with 10; NPANELPOS= 20 displays 20 odds ratios in the first panel but only one in the second. For two-way frequency plots, NPANELPOS= divides the plot into multiple panels that display at most jnj levels of the row variable per panel. The NPANELPOS= plot-option applies to two-way plots that are displayed with grouped layout, which you specify with the TWOWAY=GROUPVERTICAL or TWOWAY=GROUPHORIZONTAL plot-option. The NPANELPOS= plot-option does not apply to the TWOWAY=STACKED layout. ORDER=ASCENDING DESCENDING displays the statistics in sorted order. By default, the statistics are displayed in the order in which the corresponding strata appear in the multiway table display. The ORDER= plot-option applies to the following plots: KAPPAPLOT, ODDSRATIOPLOT, RELRISKPLOT, RISKDIFFPLOT, and WTKAPPAPLOT.

45 TABLES Statement 2251 ORIENT=HORIZONTAL VERTICAL controls the orientation of the plot. The ORIENT= plot-option applies to the following plots: CUMFREQPLOT, DEVIATIONPLOT, and FREQPLOT. ORIENT=HORIZONTAL places the variable levels on the Y axis and the frequencies, percentages, or statistic-values on the X axis. ORIENT=VERTICAL places the variable levels on the X axis. The default orientation is ORIENT=VERTICAL for bar charts (TYPE=BAR) and ORIENT=HORIZONTAL for dot plots (TYPE=DOT). RANGE= ( < min > <,max > ) CLIP specifies the range of values to display. The RANGE= plot-option applies to the following plots: KAPPAPLOT, ODDSRATIOPLOT, RELRISKPLOT, RISKDIFFPLOT, and WTKAPPAPLOT. If you specify RANGE=CLIP, the confidence limits are clipped and the display range is determined by the minimum and maximum values of the estimates. By default, the display range includes all confidence limits. SCALE=FREQ LOG PERCENT SQRT specifies the scale of the frequencies to display. The SCALE= plot-option applies to the frequency plot (FREQPLOT) and the cumulative frequency plot (CUMFREQPLOT). The default is SCALE=FREQ, which displays unscaled frequencies. SCALE=LOG displays log (base 10) frequencies. SCALE=PERCENT displays percentages (relative frequencies). SCALE=SQRT displays square roots of the frequencies, which produces a plot known as a rootogram. STATS displays the values of the statistics and their confidence limits on the right side of the plot. If you do not request the STATS plot-option, the statistic values are not displayed. The STATS plot-option applies to the following plots: KAPPAPLOT, ODDSRATIOPLOT, RELRISKPLOT, RISKDIFFPLOT, and WTKAPPAPLOT. TWOWAY=GROUPVERTICAL GROUPHORIZONTAL STACKED specifies the layout for a two-way frequency plot (FREQPLOT). The TWOWAY= plotoption applies to frequency plots for two-way and multiway table requests; PROC FREQ produces a two-way frequency plot for each stratum of a multiway table request. TWOWAY=GROUPVERTICAL produces a grouped plot with a vertical common baseline. The plot is grouped by the row variable, which is the first variable that you specify in a two-way table request. TWOWAY=GROUPHORIZONTAL produces a grouped plot with a horizontal common baseline. TWOWAY=STACKED produces stacked frequency plots for two-way tables. In a stacked bar chart, the bars correspond to the column variable values, and the row frequencies are stacked within each column. In a stacked dot plot, the dotted lines correspond to the columns, and the row frequencies within columns are plotted as data dots on the same column line. The default two-way layout is TWOWAY=GROUPVERTICAL. The ORIENT=, SCALE=, and TYPE= plot-options are available for each TWOWAY= layout.

46 2252 Chapter 35: The FREQ Procedure TYPE=BARCHART DOTPLOT specifies the plot type for frequency (FREQPLOT), cumulative frequency (CUMFREQPLOT), and deviation plots (DEVIATIONPLOT). TYPE=BARCHART produces a bar chart, and TYPE=DOTPLOT produces a dot plot. The default is TYPE=BARCHART. PRINTKWT displays the weights that PROC FREQ uses to compute the weighted kappa coefficient. You must also specify the AGREE option to request the weighted kappa coefficient. You can specify (WT=FC) with the AGREE option to request Fleiss-Cohen weights. By default, PROC FREQ uses Cicchetti-Allison weights to compute the weighted kappa coefficient. See the section Weighted Kappa Coefficient on page 2301 for details. RELRISK requests relative risk measures and their confidence limits for 2 2 tables. These measures include the odds ratio and the column 1 and 2 relative risks. See the section Odds Ratio and Relative Risks for 2 x 2 Tables on page 2294 for details. You can also obtain the RELRISK measures by specifying the MEASURES option, which produces other measures of association in addition to the relative risks. You can request exact confidence limits for the odds ratio by specifying the OR option in the EXACT statement. RISKDIFF < (riskdiff-options) > requests risks, or binomial proportions, for 2 2 tables. For column 1 and column 2, PROC FREQ computes the row 1 risk, row 2 risk, total risk, and risk difference (row 1 row 2), together with their asymptotic standard errors and asymptotic (Wald) confidence limits. PROC FREQ also provides exact (Clopper-Pearson) confidence limits for the row 1, row 2, and total risks. The ALPHA= option determines the confidence level, and the default of ALPHA=0.05 produces 95% confidence limits. See the section Risks and Risk Differences on page 2287 for details. You can specify riskdiff-options inside the parentheses following the RISKDIFF option to request tests of noninferiority, superiority, and equivalence for the risk difference. Available test methods include Farrington-Manning, Hauck-Anderson, and Newcombe score (Wilson), in addition to the Wald test. Table summarizes the riskdiff-options. You can request exact unconditional confidence limits for the risk difference by specifying the RISKDIFF option in the EXACT statement. See the section Exact Unconditional Confidence Limits for the Risk Difference on page 2293 for more information.

47 TABLES Statement 2253 Table RISKDIFF (Proportion Difference) Options Task Riskdiff-Option Specify the column COLUMN=1 2 Request a continuity correction CORRECT Request tests Specify the test method Specify the test margin Specify the test variance EQUAL EQUIV EQUIVALENCE NONINF NONINFERIORITY SUP SUPERIORITY METHOD= MARGIN= VAR=SAMPLE NULL You can specify the following riskdiff-options inside parentheses following the RISKDIFF option. COLUMN=1 2 BOTH specifies the table column for which to compute the risk difference tests of noninferiority, superiority, or equivalence, which you request with the NONINF, SUP, and EQUIV riskdiff-options, respectively. You can specify COLUMN=1, COLUMN=2, or COL- UMN=BOTH. If you do not specify the COLUMN= option, PROC FREQ computes the risk difference tests for column 1. The COLUMN= option has no effect on the table of risk estimates and confidence limits or on the equality test; PROC FREQ computes these statistics for both column 1 and column 2. CORRECT includes a continuity correction in the asymptotic Wald confidence limits and tests. The CORRECT riskdiff-option also includes a continuity correction in the Newcombe score confidence limits, which you request with the METHOD=SCORE riskdiff-option. METHOD=HA and METHOD=FM do not use continuity corrections. The CORRECT riskdiff-option has the same effect as the RISKDIFFC option. EQUAL requests a test of the null hypothesis that the risk difference equals zero. PROC FREQ provides an asymptotic Wald test of equality. See the section Equality Test on page 2289 for details. You can specify the test variance type with the VAR= riskdiffoption. EQUIV EQUIVALENCE requests a test of equivalence for the risk difference. See the section Equivalence Tests on page 2292 for details. You can specify the equivalence test margins with the MARGIN= riskdiff-option and the test method with the METHOD= riskdiff-option. PROC FREQ uses METHOD=WALD by default. MARGIN=value (lower,upper) specifies the margin for the noninferiority, superiority, and equivalence tests, which you request with the NONINF, SUP, and EQUIV riskdiff-options, respectively. If you do not specify MARGIN=, PROC FREQ uses a margin of 0.2 by default.

48 2254 Chapter 35: The FREQ Procedure For noninferiority and superiority tests, specify a single value for the MARGIN= riskdiffoption. The MARGIN= value must be a positive number. You can specify value as a number between 0 and 1. Or you can specify value in percentage form as a number between 1 and 100, and PROC FREQ converts that number to a proportion. The procedure treats the value 1 as 1%. For an equivalence test, you can specify a single MARGIN= value, or you can specify both lower and upper values. If you specify a single MARGIN= value, it must be a positive number, as described previously. If you specify a single MARGIN= value for an equivalence test, PROC FREQ uses value as the lower margin and value as the upper margin for the test. If you specify both lower and upper values for an equivalence test, you can specify them in proportion form as numbers between 1 or 1. Or you can specify them in percentage form as numbers between 100 and 100, and PROC FREQ converts the numbers to proportions. The value of lower must be less than the value of upper. METHOD=method specifies the method for the noninferiority, superiority, and equivalence tests, which you request with the NONINF, SUP, and EQUIV riskdiff-options, respectively. The following methods are available: FM HA Farrington-Manning Hauck-Anderson SCORE NEWCOMBE WILSON WALD Wald Newcombe score (Wilson) The default is METHOD=WALD. See the section Noninferiority Test on page 2289 for descriptions of these methods. For METHOD=SCORE and METHOD=WALD, you can request a continuity correction with the CORRECT riskdiff-option. For METHOD=WALD, you can specify the variance type with the VAR= riskdiff-option. NONINF NONINFERIORITY requests a test of noninferiority for the risk difference. See the section Noninferiority Test on page 2289 for details. You can specify the test margin with the MARGIN= riskdiff-option and the test method with the METHOD= riskdiff-option. PROC FREQ uses METHOD=WALD by default. SUP SUPERIORITY requests a test of superiority for the binomial proportion. See the section Superiority Test on page 2292 for details. You can specify the test margin with the MARGIN= riskdiff-option and the test method with the METHOD= riskdiff-option. PROC FREQ uses METHOD=WALD by default. VAR=SAMPLE NULL specifies the type of variance estimate to use for the Wald tests of noninferiority, superiority, equivalence, and equality. The default is VAR=SAMPLE, which estimates the variance from the sample proportions. VAR=NULL uses the null hypothesis values to compute the variance. See the sections Equality Test on page 2289 and Noninferiority Test on page 2289 for details.

49 TABLES Statement 2255 RISKDIFFC < (riskdiff-options) > requests the RISKDIFF option statistics for 2 2 tables and includes a continuity correction in the asymptotic Wald confidence limits and tests. The RISKDIFF option statistics include risks, or binomial proportions, for 2 2 tables. For column 1 and column 2, PROC FREQ computes the row 1 risk, row 2 risk, total risk, and risk difference (row 1 row 2), together with their asymptotic standard errors and asymptotic (Wald) confidence limits. PROC FREQ also provides exact (Clopper-Pearson) confidence limits for the row 1, row 2, and total risks. See the section Risks and Risk Differences on page 2287 for details. You can request additional tests and statistics for the risk difference by specifying riskdiffoptions in parentheses after RISKDIFFC. The riskdiff-options are the same as those available with RISKDIFF. See the description of the RISKDIFF option for details. You can request exact unconditional confidence limits for the risk difference by specifying the RISKDIFF option in the EXACT statement. See the section Exact Unconditional Confidence Limits for the Risk Difference on page 2293 for more information. SCORES=type specifies the type of row and column scores that PROC FREQ uses to compute the following statistics: Mantel-Haenszel chi-square, Pearson correlation, Cochran-Armitage test for trend, weighted kappa coefficient, and Cochran-Mantel-Haenszel statistics. The value of type can be one of the following: MODRIDIT RANK RIDIT TABLE See the section Scores on page 2265 for descriptions of these score types. If you do not specify the SCORES= option, PROC FREQ uses SCORES=TABLE by default. For character variables, the row and column TABLE scores are the row and column numbers. That is, the TABLE score is 1 for row 1, 2 for row 2, and so on. For numeric variables, the row and column TABLE scores equal the variable values. See the section Scores on page 2265 for details. Using MODRIDIT, RANK, or RIDIT scores yields nonparametric analyses. You can use the SCOROUT option to display the row and column scores. SCOROUT displays the row and column scores that PROC FREQ uses to compute score-based tests and statistics. You can specify the score type with the SCORES= option. See the section Scores on page 2265 for details. The scores are computed and displayed only when PROC FREQ computes statistics for twoway tables. You can use ODS to store the scores in an output data set. See the section ODS Table Names on page 2331 for more information.

50 2256 Chapter 35: The FREQ Procedure SPARSE reports all possible combinations of the variable values for an n-way table when n > 1, even if a combination does not occur in the data. The SPARSE option applies only to crosstabulation tables displayed in LIST format and to the OUT= output data set. If you do not use the LIST or OUT= option, the SPARSE option has no effect. When you specify the SPARSE and LIST options, PROC FREQ displays all combinations of variable values in the table listing, including those with a frequency count of zero. By default, without the SPARSE option, PROC FREQ does not display zero-frequency levels in LIST output. When you use the SPARSE and OUT= options, PROC FREQ includes empty crosstabulation table cells in the output data set. By default, PROC FREQ does not include zero-frequency table cells in the output data set. See the section Missing Values on page 2261 for more information. TESTF=(values) specifies the null hypothesis frequencies for a one-way chi-square goodness-of-fit test, which you request with the CHISQ option. See the section Chi-Square Test for One-Way Tables on page 2266 for details. You can separate the TESTF= values with blanks or commas. The number of values must equal the number of variable levels in the one-way table. The sum of the values must equal the total frequency for the one-way table. List the values in the order in which the corresponding variable levels appear in the output. If you omit the CHISQ option, the TESTF= option invokes CHISQ. TESTP=(values) specifies the null hypothesis proportions for a one-way chi-square goodness-of-fit test, which you request with the CHISQ option. See the section Chi-Square Test for One-Way Tables on page 2266 for details. You can separate the TESTP= values with blanks or commas. The number of values must equal the number of variable levels in the one-way table. List the values in the order in which the corresponding variable levels appear in the output. You can specify values in probability form as numbers between 0 and 1, where the proportions sum to 1. Or you can specify values in percentage form as numbers between 0 and 100, where the percentages sum to 100. If you omit the CHISQ option, the TESTP= option invokes CHISQ. TOTPCT displays the percentage of the total multiway table frequency in crosstabulation tables for n-way tables, where n > 2. By default, PROC FREQ displays the percentage of the individual two-way table frequency but does not display the percentage of the total frequency for multiway crosstabulation tables. See the section Multiway Tables on page 2324 for more information. TREND The percentage of total multiway table frequency is displayed by default when you specify the LIST option. It is also provided by default in the PERCENT variable in the OUT= output data set. requests the Cochran-Armitage test for trend. The table must be 2 C or R 2 to compute the trend test. See the section Cochran-Armitage Test for Trend on page 2296 for details.

51 TEST Statement 2257 TEST Statement TEST options ; The TEST statement requests asymptotic tests for measures of association and measures of agreement. You must use a TABLES statement with the TEST statement. options specify the statistics for which to provide asymptotic tests. Table lists the available statistics, which include measures of association and agreement. The option names are identical to those in the TABLES and OUTPUT statements. You can request all tests for groups of statistics by using group options MEASURES or AGREE. Or you can request tests individually by using the options shown in Table For each measure of association or agreement that you specify, PROC FREQ provides an asymptotic test that the measure equals zero. PROC FREQ displays the asymptotic standard error under the null hypothesis, the test statistic, and the p-values. Additionally, PROC FREQ reports the confidence limits for the measure. The ALPHA= option in the TABLES statement determines the confidence level, which by default equals 0.05 and provides 95% confidence limits. See the sections Asymptotic Tests on page 2271 and Confidence Limits on page 2271 for details. Also see the section Statistical Computations on page 2264 for information about individual measures. You can request exact tests for selected measures of association and agreement by using the EXACT statement. See the section Exact Statistics on page 2314 for more information. If you use only one TABLES statement, you do not need to specify the same options in both the TABLES and TEST statements; when you specify an option in the TEST statement, PROC FREQ automatically invokes the corresponding TABLES statement option. However, when you use the TEST statement with multiple TABLES statements, you must specify options in the TABLES statements to request the desired statistics. PROC FREQ then provides asymptotic tests for those statistics that you also specify in the TEST statement. Table TEST Statement Options Option Asymptotic Tests Computed Required TABLES Statement Option AGREE simple and weighted kappa coefficients AGREE GAMMA gamma ALL or MEASURES KAPPA simple kappa coefficient AGREE KENTB Kendall s tau-b ALL or MEASURES MEASURES gamma, Kendall s tau-b, Stuart s tau-c, ALL or MEASURES Somers D.C jr/, Somers D.RjC /, Pearson and Spearman correlations PCORR Pearson correlation coefficient ALL or MEASURES SCORR Spearman correlation coefficient ALL or MEASURES SMDCR Somers D.C jr/ ALL or MEASURES SMDRC Somers D.RjC / ALL or MEASURES STUTC Stuart s tau-c ALL or MEASURES WTKAP weighted kappa coefficient AGREE

52 2258 Chapter 35: The FREQ Procedure WEIGHT Statement WEIGHT variable < / option > ; The WEIGHT statement names a numeric variable that provides a weight for each observation in the input data set. The WEIGHT statement is most commonly used to input cell count data. See the section Inputting Frequency Counts on page 2259 for more information. If you use a WEIGHT statement, PROC FREQ assumes that an observation represents n observations, where n is the value of variable. The value of the WEIGHT variable is not required to be an integer. If the value of the WEIGHT variable is missing, PROC FREQ does not use that observation in the analysis. If the value of the WEIGHT variable is zero, PROC FREQ ignores the observation unless you specify the ZEROS option, which includes observations with zero weights. If you do not specify a WEIGHT statement, each observation has a default weight of 1. The sum of the WEIGHT variable values represents the total number of observations. If any value of the WEIGHT variable is negative, PROC FREQ displays the frequencies computed from the weighted values but does not compute percentages and statistics. If you create an output data set by using the OUT= option in the TABLES statement, PROC FREQ assigns missing values to the PERCENT variable. PROC FREQ also assigns missing values to the variables that the OUTEXPECT and OUTPCT options provide. If any value of the WEIGHT variable is negative, you cannot create an output data set by using the OUTPUT statement because statistics are not computed when there are negative weights. You can specify the following option in the WEIGHT statement: ZEROS includes observations with zero weight values. By default, PROC FREQ ignores observations with zero weights. If you specify the ZEROS option, frequency and and crosstabulation tables display any levels corresponding to observations with zero weights. Without the ZEROS option, PROC FREQ does not process observations with zero weights, and so does not display levels that contain only observations with zero weights. With the ZEROS option, PROC FREQ includes levels with zero weights in the chi-square goodness-of-fit test for one-way tables. Also, PROC FREQ includes any levels with zero weights in binomial computations for one-way tables. This makes it possible to compute binomial tests and estimates when the specified level contains no observations with positive weights. For two-way tables, the ZEROS option enables computation of kappa statistics when there are levels that contain no observations with positive weight. For more information, see the section Tables with Zero Rows and Columns on page Note that even with the ZEROS option, PROC FREQ does not compute the CHISQ or MEASURES statistics for two-way tables when the table has a zero row or zero column because most of these statistics are undefined in this case. The ZEROS option invokes the SPARSE option in the TABLES statement, which includes

53 Details: FREQ Procedure 2259 table cells with a zero frequency count in the LIST output and in the OUT= data set. By default, without the SPARSE option, PROC FREQ does not include zero frequency cells in the LIST output or in the OUT= data set. If you specify the ZEROS option in the WEIGHT statement but do not want the SPARSE option, you can specify the NOSPARSE option in the TABLES statement. Details: FREQ Procedure Inputting Frequency Counts PROC FREQ can use either raw data or cell count data to produce frequency and crosstabulation tables. Raw data, also known as case-record data, report the data as one record for each subject or sample member. Cell count data report the data as a table, listing all possible combinations of data values along with the frequency counts. This way of presenting data often appears in published results. The following DATA step statements store raw data in a SAS data set: data Raw; input Subject $ R datalines; ; You can store the same data as cell counts by using the following DATA step statements: data CellCounts; input R C datalines; ; The variable R contains the values for the rows, and the variable C contains the values for the columns. The variable Count contains the cell count for each row and column combination. Both the Raw data set and the CellCounts data set produce identical frequency counts, two-way tables, and statistics. When using the CellCounts data set, you must include a WEIGHT statement to specify that the variable Count contains cell counts. For example, the following PROC FREQ statements create a two-way crosstabulation table by using the CellCounts data set: proc freq data=cellcounts; tables R*C; weight Count; run;

54 2260 Chapter 35: The FREQ Procedure Grouping with Formats PROC FREQ groups a variable s values according to its formatted values. If you assign a format to a variable with a FORMAT statement, PROC FREQ formats the variable values before dividing observations into the levels of a frequency or crosstabulation table. For example, suppose that variable X has the values 1.1, 1.4, 1.7, 2.1, and 2.3. Each of these values appears as a level in the frequency table. If you decide to round each value to a single digit, include the following statement in the PROC FREQ step: format X 1.; Now the table lists the frequency count for formatted level 1 as two and for formatted level 2 as three. PROC FREQ treats formatted character variables in the same way. The formatted values are used to group the observations into the levels of a frequency table or crosstabulation table. PROC FREQ uses the entire value of a character format to classify an observation. You can also use the FORMAT statement to assign formats that were created with the FORMAT procedure to the variables. User-written formats determine the number of levels for a variable and provide labels for a table. If you use the same data with different formats, then you can produce frequency counts and statistics for different classifications of the variable values. When you use PROC FORMAT to create a user-written format that combines missing and nonmissing values into one category, PROC FREQ treats the entire category of formatted values as missing. For example, a questionnaire codes 1 as yes, 2 as no, and 8 as a no answer. The following PROC FORMAT statements create a user-written format: proc format; value Questfmt 1 ='Yes' 2 ='No' 8,. ='Missing'; run; When you use a FORMAT statement to assign Questfmt. to a variable, the variable s frequency table no longer includes a frequency count for the response of 8. You must use the MISSING or MISSPRINT option in the TABLES statement to list the frequency for no answer. The frequency count for this level includes observations with either a value of 8 or a missing value (.). The frequency or crosstabulation table lists the values of both character and numeric variables in ascending order based on internal (unformatted) variable values unless you change the order with the ORDER= option. To list the values in ascending order by formatted values, use OR- DER=FORMATTED in the PROC FREQ statement. For more information about the FORMAT statement, see SAS Language Reference: Concepts.

55 Missing Values 2261 Missing Values When the value of the WEIGHT variable is missing, PROC FREQ does not include that observation in the analysis. PROC FREQ treats missing BY variable values like any other BY variable value. The missing values form a separate BY group. If an observation has a missing value for a variable in a TABLES request, by default PROC FREQ does not include that observation in the frequency or crosstabulation table. Also by default, PROC FREQ does not include observations with missing values in the computation of percentages and statistics. The procedure displays the number of missing observations below each table. PROC FREQ also reports the number of missing values in output data sets. The TABLES statement OUT= data set includes an observation that contains the missing value frequency. The NMISS option in the OUTPUT statement provides an output data set variable that contains the missing value frequency. The following options change the way in which PROC FREQ handles missing values of TABLES variables: MISSPRINT MISSING displays missing value frequencies in frequency or crosstabulation tables but does not include them in computations of percentages or statistics. treats missing values as a valid nonmissing level for all TABLES variables. Displays missing levels in frequency and crosstabulation tables and includes them in computations of percentages and statistics. This example shows the three ways that PROC FREQ can handle missing values of TABLES variables. The following DATA step statements create a data set with a missing value for the variable A: data one; input A Freq; datalines; ; The following PROC FREQ statements request a one-way frequency table for the variable A. The first request does not specify a missing value option. The second request specifies the MISSPRINT option in the TABLES statement. The third request specifies the MISSING option in the TABLES statement. proc freq data=one; tables A; weight Freq; title 'Default'; run;

56 2262 Chapter 35: The FREQ Procedure proc freq data=one; tables A / missprint; weight Freq; title 'MISSPRINT Option'; run; proc freq data=one; tables A / missing; weight Freq; title 'MISSING Option'; run; Figure displays the frequency tables produced by this example. The first table shows PROC FREQ s default behavior for handling missing values. The observation with a missing value of the TABLES variable A is not included in the table, and the frequency of missing values is displayed below the table. The second table, for which the MISSPRINT option is specified, displays the missing observation but does not include its frequency when computing the total frequency and percentages. The third table shows that PROC FREQ treats the missing level as a valid nonmissing level when the MISSING option is specified. The table displays the missing level, and PROC FREQ includes this level when computing frequencies and percentages. Figure Missing Values in Frequency Tables Default The FREQ Procedure Cumulative Cumulative A Frequency Percent Frequency Percent Frequency Missing = 2 MISSPRINT Option The FREQ Procedure Cumulative Cumulative A Frequency Percent Frequency Percent Frequency Missing = 2 MISSING Option The FREQ Procedure Cumulative Cumulative A Frequency Percent Frequency Percent

57 In-Database Computation 2263 When a combination of variable values for a two-way table is missing, PROC FREQ assigns zero to the frequency count for the table cell. By default, PROC FREQ does not display missing combinations in LIST format. Also, PROC FREQ does not include missing combinations in the OUT= output data set by default. To include missing combinations, you can specify the SPARSE option with the LIST or OUT= option in the TABLES statement. In-Database Computation The FREQ procedure can use in-database computation to construct frequency and crosstabulation tables when the DATA= input data set is stored as a table in a supported database management system (DBMS). Supported databases include Teradata, DB2 under UNIX, and Oracle. In-database computation can provide the advantages of faster processing and reduced data transfer between the database and SAS software. For information about in-database computation, see the section In-Database Procedures in SAS/ACCESS 9.2 for Relational Databases: Reference. PROC FREQ performs in-database computation by using SQL implicit pass-through. The procedure generates SQL queries that are based on the tables that you request in the TABLES statement. The database executes these SQL queries to construct initial summary tables, which are then transmitted to PROC FREQ. The procedure uses this summary information to perform the remaining analyses and tasks in the usual way (out of the database). So instead of transferring the entire data set over the network between the database and SAS software, the in-database method transfers only the summary tables. This can substantially reduce processing time when the dimensions of the summary tables (in terms of rows and columns) are much smaller than the dimensions of the entire database table (in terms of individual observations). Additionally, in-database summarization uses efficient parallel processing, which can also provide performance advantages. In-database computation is controlled by the SQLGENERATION option, which you can specify in either a LIBNAME statement or an OPTIONS statement. See the section In-Database Procedures in SAS/ACCESS 9.2 for Relational Databases: Reference for details about the SQLGENERATION option and other options that affect in-database computation. By default, PROC FREQ uses indatabase computation when possible. There are no FREQ procedure options that control in-database computation. PROC FREQ uses formatted values to group observations into the levels of frequency and crosstabulation tables. See the section Grouping with Formats on page 2260 for more information. If formats are available in the database, then in-database summarization uses the formats. If formats are not available in the database, then in-database summarization is based on the raw data values, and PROC FREQ performs the final, formatted classification (out of the database). For more information, see the section Deploying and Using SAS Formats in Teradata in SAS/ACCESS 9.2 for Relational Databases: Reference. The order of observations is not inherently defined for DBMS tables. The following options relate to the order of observations and therefore should not be specified for PROC FREQ in-database computation: If you specify the FIRSTOBS= or OBS= data set option, PROC FREQ does not perform in-database computation.

58 2264 Chapter 35: The FREQ Procedure If you specify the NOTSORTED option in the BY statement, PROC FREQ in-database computation ignores it and uses the default ASCENDING order for BY variables. If you specify the ORDER=DATA option for input data in a DBMS table, PROC FREQ computation might produce different results for separate runs of the same analysis. In addition to determining the order of variable levels in crosstabulation table displays, the ORDER= option can also affect the values of many of the test statistics and measures that PROC FREQ computes. Statistical Computations Definitions and Notation A two-way table represents the crosstabulation of row variable X and column variable Y. Let the table row values or levels be denoted by X i, i D 1; 2; : : : ; R, and the column values by Y j, j D 1; 2; : : : ; C. Let n ij denote the frequency of the table cell in the ith row and j th column and define the following notation: n i D X j n ij (row totals) n j D X i n D X i n ij X j p ij D n ij =n p i D n i =n p j D n j =n n ij (column totals) (overall total) (cell percentages) (row percentages of total) (column percentages of total) R i D score for row i C j D score for column j NR D X n i R i =n (average row score) i NC D X j n j C j =n (average column score) A ij D X X n kl C X X n kl k>i l>j k<i l<j D ij D X X n kl C X X n kl k>i l<j k<i l>j P D X X n ij A ij (twice the number of concordances) i j Q D X X n ij D ij (twice the number of discordances) i j

59 Statistical Computations 2265 Scores PROC FREQ uses scores of the variable values to compute the Mantel-Haenszel chi-square, Pearson correlation, Cochran-Armitage test for trend, weighted kappa coefficient, and Cochran-Mantel- Haenszel statistics. The SCORES= option in the TABLES statement specifies the score type that PROC FREQ uses. The available score types are TABLE, RANK, RIDIT, and MODRIDIT scores. The default score type is TABLE. Using MODRIDIT, RANK, or RIDIT scores yields nonparametric analyses. For numeric variables, table scores are the values of the row and column levels. If the row or column variable is formatted, then the table score is the internal numeric value corresponding to that level. If two or more numeric values are classified into the same formatted level, then the internal numeric value for that level is the smallest of these values. For character variables, table scores are defined as the row numbers and column numbers (that is, 1 for the first row, 2 for the second row, and so on). Rank scores, which you request with the SCORES=RANK option, are defined as R1 i D X k<i n k C.n i C 1/=2 i D 1; 2; : : : ; R C1 j D X l<j n l C.n j C 1/=2 j D 1; 2; : : : ; C where R1 i is the rank score of row i, and C1 j is the rank score of column j. Note that rank scores yield midranks for tied values. Ridit scores, which you request with the SCORES=RIDIT option, are defined as rank scores standardized by the sample size (Bross 1958, Mack and Skillings 1980). Ridit scores are derived from the rank scores as R2 i D R1 i =n C 2 j D C1 j =n i D 1; 2; : : : ; R j D 1; 2; : : : ; C Modified ridit scores (SCORES=MODRIDIT) represent the expected values of the order statistics of the uniform distribution on (0,1) (van Elteren 1960, Lehmann 1975). Modified ridit scores are derived from rank scores as R3 i D R1 i =.n C 1/ C 3 j D C1 j =.n C 1/ i D 1; 2; : : : ; R j D 1; 2; : : : ; C Chi-Square Tests and Statistics The CHISQ option provides chi-square tests of homogeneity or independence and measures of association based on the chi-square statistic. When you specify the CHISQ option in the TABLES statement, PROC FREQ computes the following chi-square tests for each two-way table: the Pearson chi-square, likelihood-ratio chi-square, and Mantel-Haenszel chi-square. PROC FREQ provides the following measures of association based on the Pearson chi-square statistic: the phi coefficient, contingency coefficient, and Cramer s V. For 2 2 tables, the CHISQ option also provides Fisher s exact test and the continuity-adjusted chi-square. You can request Fisher s exact test for general R C tables by specifying the FISHER option in the TABLES or EXACT statement.

60 2266 Chapter 35: The FREQ Procedure For one-way frequency tables, the CHISQ option provides a chi-square goodness-of-fit test. The other chi-square tests and statistics described in this section are computed only for two-way tables. All of the two-way test statistics described in this section test the null hypothesis of no association between the row variable and the column variable. When the sample size n is large, these test statistics have an asymptotic chi-square distribution when the null hypothesis is true. When the sample size is not large, exact tests might be useful. PROC FREQ provides exact tests for the Pearson chi-square, the likelihood-ratio chi-square, and the Mantel-Haenszel chi-square (in addition to Fisher s exact test). PROC FREQ also provides an exact chi-square goodness-of-fit test for one-way tables. You can request these exact tests by specifying the corresponding options in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Note that the Mantel-Haenszel chi-square statistic is appropriate only when both variables lie on an ordinal scale. The other chi-square tests and statistics in this section are appropriate for either nominal or ordinal variables. The following sections give the formulas that PROC FREQ uses to compute the chi-square tests and statistics. See Agresti (2007), Stokes, Davis, and Koch (2000), and the other references cited for each statistic for more information. Chi-Square Test for One-Way Tables For one-way frequency tables, the CHISQ option in the TABLES statement provides a chi-square goodness-of-fit test. Let C denote the number of classes, or levels, in the one-way table. Let f i denote the frequency of class i (or the number of observations in class i) for i D 1; 2; : : : ; C. Then PROC FREQ computes the one-way chi-square statistic as Q P D CX.f i e i / 2 id1 e i where e i is the expected frequency for class i under the null hypothesis. In the test for equal proportions, which is the default for the CHISQ option, the null hypothesis specifies equal proportions of the total sample size for each class. Under this null hypothesis, the expected frequency for each class equals the total sample size divided by the number of classes, e i D n=c for i D 1; 2; : : : ; C In the test for specified frequencies, which PROC FREQ computes when you input null hypothesis frequencies by using the TESTF= option, the expected frequencies are the TESTF= values that you specify. In the test for specified proportions, which PROC FREQ computes when you input null hypothesis proportions by using the TESTP= option, the expected frequencies are determined from the specified TESTP= proportions p i as e i D p i n for i D 1; 2; : : : ; C Under the null hypothesis (of equal proportions, specified frequencies, or specified proportions), Q P has an asymptotic chi-square distribution with C 1 degrees of freedom. In addition to the asymptotic test, you can request an exact one-way chi-square test by specifying the CHISQ option in the EXACT statement. See the section Exact Statistics on page 2314 for more information.

61 Statistical Computations 2267 Pearson Chi-Square Test for Two-Way Tables The Pearson chi-square for two-way tables involves the differences between the observed and expected frequencies, where the expected frequencies are computed under the null hypothesis of independence. The Pearson chi-square statistic is computed as X.n ij e ij / 2 Q P D X i j e ij where n ij is the observed frequency in table cell (i; j ) and e ij is the expected frequency for table cell (i; j ). The expected frequency is computed under the null hypothesis that the row and column variables are independent, e ij D n i n j n When the row and column variables are independent, Q P has an asymptotic chi-square distribution with.r 1/.C 1/ degrees of freedom. For large values of Q P, this test rejects the null hypothesis in favor of the alternative hypothesis of general association. In addition to the asymptotic test, you can request an exact Pearson chi-square test by specifying the PCHI or CHISQ option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. For 2 2 tables, the Pearson chi-square is also appropriate for testing the equality of two binomial proportions. For R 2 and 2 C tables, the Pearson chi-square tests the homogeneity of proportions. See Fienberg (1980) for details. Likelihood-Ratio Chi-Square Test The likelihood-ratio chi-square involves the ratios between the observed and expected frequencies. The likelihood-ratio chi-square statistic is computed as G 2 D 2 X X nij n ij ln e ij i j where n ij is the observed frequency in table cell (i; j ) and e ij is the expected frequency for table cell (i; j ). When the row and column variables are independent, G 2 has an asymptotic chi-square distribution with.r 1/.C 1/ degrees of freedom. In addition to the asymptotic test, you can request an exact likelihood-ratio chi-square test by specifying the LRCHI or CHISQ option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Continuity-Adjusted Chi-Square Test The continuity-adjusted chi-square for 2 2 tables is similar to the Pearson chi-square, but it is adjusted for the continuity of the chi-square distribution. The continuity-adjusted chi-square is most

62 2268 Chapter 35: The FREQ Procedure useful for small sample sizes. The use of the continuity adjustment is somewhat controversial; this chi-square test is more conservative (and more like Fisher s exact test) when the sample size is small. As the sample size increases, the continuity-adjusted chi-square becomes more like the Pearson chi-square. The continuity-adjusted chi-square statistic is computed as Q C D X i X j max.0; jn ij e ij j 0:5/ 2 e ij Under the null hypothesis of independence, Q C has an asymptotic chi-square distribution with.r 1/.C 1/ degrees of freedom. Mantel-Haenszel Chi-Square Test The Mantel-Haenszel chi-square statistic tests the alternative hypothesis that there is a linear association between the row variable and the column variable. Both variables must lie on an ordinal scale. The Mantel-Haenszel chi-square statistic is computed as Q MH D.n 1/r 2 where r 2 is the Pearson correlation between the row variable and the column variable. For a description of the Pearson correlation, see the Pearson Correlation Coefficient on page The Pearson correlation and thus the Mantel-Haenszel chi-square statistic use the scores that you specify in the SCORES= option in the TABLES statement. See Mantel and Haenszel (1959) and Landis, Heyman, and Koch (1978) for more information. Under the null hypothesis of no association, Q MH has an asymptotic chi-square distribution with one degree of freedom. In addition to the asymptotic test, you can request an exact Mantel-Haenszel chi-square test by specifying the MHCHI or CHISQ option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Fisher s Exact Test Fisher s exact test is another test of association between the row and column variables. This test assumes that the row and column totals are fixed, and then uses the hypergeometric distribution to compute probabilities of possible tables conditional on the observed row and column totals. Fisher s exact test does not depend on any large-sample distribution assumptions, and so it is appropriate even for small sample sizes and for sparse tables. 2 2 Tables For 2 2 tables, PROC FREQ gives the following information for Fisher s exact test: table probability, two-sided p-value, left-sided p-value, and right-sided p-value. The table probability equals the hypergeometric probability of the observed table, and is in fact the value of the test statistic for Fisher s exact test.

63 Statistical Computations 2269 Where p is the hypergeometric probability of a specific table with the observed row and column totals, Fisher s exact p-values are computed by summing probabilities p over defined sets of tables, PROB D X A p The two-sided p-value is the sum of all possible table probabilties (conditional on the observed row and column totals) that are less than or equal to the observed table probability. For the two-sided p-value, the set A includes all possible tables with hypergeometric probabilities less than or equal to the probability of the observed table. A small two-sided p-value supports the alternative hypothesis of association between the row and column variables. For 2 2 tables, one-sided p-values for Fisher s exact test are defined in terms of the frequency of the cell in the first row and first column of the table, the (1,1) cell. Denoting the observed (1,1) cell frequency by n 11, the left-sided p-value for Fisher s exact test is the probability that the (1,1) cell frequency is less than or equal to n 11. For the left-sided p-value, the set A includes those tables with a (1,1) cell frequency less than or equal to n 11. A small left-sided p-value supports the alternative hypothesis that the probability of an observation being in the first cell is actually less than expected under the null hypothesis of independent row and column variables. Similarly, for a right-sided alternative hypothesis, A is the set of tables where the frequency of the (1,1) cell is greater than or equal to that in the observed table. A small right-sided p-value supports the alternative that the probability of the first cell is actually greater than that expected under the null hypothesis. Because the (1,1) cell frequency completely determines the 2 2 table when the marginal row and column sums are fixed, these one-sided alternatives can be stated equivalently in terms of other cell probabilities or ratios of cell probabilities. The left-sided alternative is equivalent to an odds ratio less than 1, where the odds ratio equals (n 11 n 22 =n 12 n 21 ). Additionally, the left-sided alternative is equivalent to the column 1 risk for row 1 being less than the column 1 risk for row 2, p 1j1 < p 1j2. Similarly, the right-sided alternative is equivalent to the column 1 risk for row 1 being greater than the column 1 risk for row 2, p 1j1 > p 1j2. See Agresti (2007) for details. R C Tables Fisher s exact test was extended to general R C tables by Freeman and Halton (1951), and this test is also known as the Freeman-Halton test. For R C tables, the two-sided p-value definition is the same as for 2 2 tables. The set A contains all tables with p less than or equal to the probability of the observed table. A small p-value supports the alternative hypothesis of association between the row and column variables. For R C tables, Fisher s exact test is inherently two-sided. The alternative hypothesis is defined only in terms of general, and not linear, association. Therefore, Fisher s exact test does not have right-sided or left-sided p-values for general R C tables. For R C tables, PROC FREQ computes Fisher s exact test by using the network algorithm of Mehta and Patel (1983), which provides a faster and more efficient solution than direct enumeration. See the section Exact Statistics on page 2314 for more details. Phi Coefficient The phi coefficient is a measure of association derived from the Pearson chi-square. The range of the phi coefficient is 1 1 for 2 2 tables. For tables larger than 2 2, the range is

64 2270 Chapter 35: The FREQ Procedure 0 min. p R 1; p C 1/ (Liebetrau 1983). The phi coefficient is computed as D.n 11 n 22 n 12 n 21 / = p n 1 n 2 n 1 n 2 for 2 2 tables D p Q P =n otherwise See Fleiss, Levin, and Paik (2003, pp ) for more information. Contingency Coefficient The contingency coefficient is a measure of association derived from the Pearson chi-square. The range of the contingency coefficient is 0 P p.m 1/=m, where m D min.r; C / (Liebetrau 1983). The contingency coefficient is computed as P D p Q P =.Q P C n/ See Kendall and Stuart (1979, pp ) for more information. Cramer s V Cramer s V is a measure of association derived from the Pearson chi-square. It is designed so that the attainable upper bound is always 1. The range of Cramer s V is 1 V 1 for 2 2 tables; for tables larger than 2 2, the range is 0 V 1. Cramer s V is computed as V D for 2 2tables s Q P =n V D min.r 1; C 1/ otherwise See Kendall and Stuart (1979, p. 588) for more information. Measures of Association When you specify the MEASURES option in the TABLES statement, PROC FREQ computes several statistics that describe the association between the row and column variables of the contingency table. The following are measures of ordinal association that consider whether the column variable Y tends to increase as the row variable X increases: gamma, Kendall s tau-b, Stuart s tau-c, and Somers D. These measures are appropriate for ordinal variables, and they classify pairs of observations as concordant or discordant. A pair is concordant if the observation with the larger value of X also has the larger value of Y. A pair is discordant if the observation with the larger value of X has the smaller value of Y. See Agresti (2007) and the other references cited for the individual measures of association. The Pearson correlation coefficient and the Spearman rank correlation coefficient are also appropriate for ordinal variables. The Pearson correlation describes the strength of the linear association between the row and column variables, and it is computed by using the row and column scores specified by the SCORES= option in the TABLES statement. The Spearman correlation is computed with rank scores. The polychoric correlation (requested by the PLCORR option) also requires ordinal variables and assumes that the variables have an underlying bivariate normal distribution. The following

65 Statistical Computations 2271 measures of association do not require ordinal variables and are appropriate for nominal variables: lambda asymmetric, lambda symmetric, and the uncertainty coefficients. PROC FREQ computes estimates of the measures according to the formulas given in the following sections. For each measure, PROC FREQ computes an asymptotic standard error (ASE), which is the square root of the asymptotic variance denoted by var in the following sections. Confidence Limits If you specify the CL option in the TABLES statement, PROC FREQ computes asymptotic confidence limits for all MEASURES statistics. The confidence coefficient is determined according to the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits. The confidence limits are computed as est. z =2 ASE / where est is the estimate of the measure, z =2 is the =2/th percentile of the standard normal distribution, and ASE is the asymptotic standard error of the estimate. Asymptotic Tests For each measure that you specify in the TEST statement, PROC FREQ computes an asymptotic test of the null hypothesis that the measure equals zero. Asymptotic tests are available for the following measures of association: gamma, Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, Somers D.RjC /, the Pearson correlation coefficient, and the Spearman rank correlation coefficient. To compute an asymptotic test, PROC FREQ uses a standardized test statistic z, which has an asymptotic standard normal distribution under the null hypothesis. The test statistic is computed as z D est = p var 0.est/ where est is the estimate of the measure and var 0.est/ is the variance of the estimate under the null hypothesis. Formulas for var 0.est/ for the individual measures of association are given in the following sections. Note that the ratio of est to p var 0.est/ is the same for the following measures: gamma, Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, and Somers D.RjC /. Therefore, the tests for these measures are identical. For example, the p-values for the test of H 0 W gamma D 0 equal the p-values for the test of H 0 W tau-b D 0. PROC FREQ computes one-sided and two-sided p-values for each of these tests. When the test statistic z is greater than its null hypothesis expected value of zero, PROC FREQ displays the right-sided p-value, which is the probability of a larger value of the statistic occurring under the null hypothesis. A small right-sided p-value supports the alternative hypothesis that the true value of the measure is greater than zero. When the test statistic is less than or equal to zero, PROC FREQ displays the left-sided p-value, which is the probability of a smaller value of the statistic occurring under the null hypothesis. A small left-sided p-value supports the alternative hypothesis that the true value of the measure is less than zero. The one-sided p-value P 1 can be expressed as ( Prob.Z > z/ if z > 0 P 1 D Prob.Z < z/ if z 0

66 2272 Chapter 35: The FREQ Procedure where Z has a standard normal distribution. The two-sided p-value P 2 is computed as P 2 D Prob.jZj > jzj/ Exact Tests Exact tests are available for the following measures of association: Kendall s tau-b, Stuart s tau-c, Somers D.C jr/ and.rjc /, the Pearson correlation coefficient, and the Spearman rank correlation coefficient. If you request an exact test for a measure of association in the EXACT statement, PROC FREQ computes the exact test of the hypothesis that the measure equals zero. See the section Exact Statistics on page 2314 for details. Gamma The gamma ( ) statistic is based only on the number of concordant and discordant pairs of observations. It ignores tied pairs (that is, pairs of observations that have equal values of X or equal values of Y ). Gamma is appropriate only when both variables lie on an ordinal scale. The range of gamma is 1 1. If the row and column variables are independent, then gamma tends to be close to zero. Gamma is estimated by G D.P Q/ =.P C Q/ and the asymptotic variance is 16 X X var.g/ D.P C Q/ 4 n ij.qa ij PD ij / 2 i j For 2 2 tables, gamma is equivalent to Yule s Q. See Goodman and Kruskal (1979) and Agresti (2002) for more information. The variance under the null hypothesis that gamma equals zero is computed as var 0.G/ D 0 X X.P C Q/ 2 n ij.a ij i j D ij / 2.P 1 Q/ 2 =na See Brown and Benedetti (1977) for details. Kendall s Tau-b Kendall s tau-b ( b ) is similar to gamma except that tau-b uses a correction for ties. Tau-b is appropriate only when both variables lie on an ordinal scale. The range of tau-b is 1 b 1. Kendall s tau-b is estimated by t b D.P Q/ = p w r w c and the asymptotic variance is 0 1 var.t b / D X X w 4 n ij.2wd ij C t b v ij / 2 n 3 t 2 b.w r C w c / 2 A i j

67 Statistical Computations 2273 where w D p w r w c X w r D n 2 w c D n 2 X i j n 2 i n 2 j d ij D A ij D ij v ij D n i w c C n j w r See Kendall (1955) for more information. The variance under the null hypothesis that tau-b equals zero is computed as 0 var 0.t b / D X X n ij.a ij w r w c i j D ij / 2.P 1 Q/ 2 =na See Brown and Benedetti (1977) for details. PROC FREQ also provides an exact test for the Kendall s tau-b. You can request this test by specifying the KENTB option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Stuart s Tau-c Stuart s tau-c ( c ) makes an adjustment for table size in addition to a correction for ties. Tau-c is appropriate only when both variables lie on an ordinal scale. The range of tau-c is 1 c 1. Stuart s tau-c is estimated by t c D m.p Q/ = n 2.m 1/ and the asymptotic variance is 0 4m 2 var.t c / X.m 1/ 2 n 4 i 1 X n ij dij 2.P Q/ 2 =na j where m D min.r; C / and d ij D A ij D ij. The variance under the null hypothesis that tau-c equals zero is the same as the asymptotic variance var, var 0.t c / D var.t c / See Brown and Benedetti (1977) for details. PROC FREQ also provides an exact test for the Stuart s tau-c. You can request this test by specifying the STUTC option in the EXACT statement. See the section Exact Statistics on page 2314 for more information.

68 2274 Chapter 35: The FREQ Procedure Somers D Somers D.C jr/ and Somers D.RjC / are asymmetric modifications of tau-b. C jr indicates that the row variable X is regarded as the independent variable and the column variable Y is regarded as dependent. Similarly, RjC indicates that the column variable Y is regarded as the independent variable and the row variable X is regarded as dependent. Somers D differs from tau-b in that it uses a correction only for pairs that are tied on the independent variable. Somers D is appropriate only when both variables lie on an ordinal scale. The range of Somers D is 1 D 1. Somers D.C jr/ is computed as D.C jr/ D.P Q/ = w r and its asymptotic variance is var.d.c jr// D 4 w 4 r X X n ij w r d ij.p Q/.n n i / 2 i j where d ij D A ij w r D n 2 D ij and X i n 2 i See Somers (1962), Goodman and Kruskal (1979), and Liebetrau (1983) for more information. The variance under the null hypothesis that D.C jr/ equals zero is computed as 0 var 0.D.C jr// D X X wr 2 n ij.a ij i j D ij / 2.P 1 Q/ 2 =na See Brown and Benedetti (1977) for details. Formulas for Somers D.RjC / are obtained by interchanging the indices. PROC FREQ also provides exact tests for Somers D.C jr/ and.rjc /. You can request these tests by specifying the SMDCR and SMDCR options in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Pearson Correlation Coefficient The Pearson correlation coefficient () is computed by using the scores specified in the SCORES= option. This measure is appropriate only when both variables lie on an ordinal scale. The range of the Pearson correlation is 1 1. The Pearson correlation coefficient is estimated by r D v=w D ss rc = p ss r ss c and its asymptotic variance is var.r/ D 1 X X w 4 n ij w.r i NR/.C j NC / i j b ij v 2 2w

69 Statistical Computations 2275 where R i and C j are the row and column scores and ss r D X i ss c D X i ss rc D X i X n ij.r i NR/ 2 j X n ij.c j NC / 2 j X n ij.r i NR/.C j NC / j b ij D.R i NR/ 2 ss c C.C j NC / 2 ss r v D ss rc w D p ss r ss c See Snedecor and Cochran (1989) for more information. The SCORES= option in the TABLES statement determines the type of row and column scores used to compute the Pearson correlation (and other score-based statistics). The default is SCORES=TABLE. See the section Scores on page 2265 for details about the available score types and how they are computed. The variance under the null hypothesis that the correlation equals zero is computed as 0 1 var 0.r/ X X n ij.r i NR/ 2.C j NC / 2 ssrc 2 =n A = ss r ss c i j Note that this expression for the variance is derived for multinomial sampling in a contingency table framework, and it differs from the form obtained under the assumption that both variables are continuous and normally distributed. See Brown and Benedetti (1977) for details. PROC FREQ also provides an exact test for the Pearson correlation coefficient. You can request this test by specifying the PCORR option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Spearman Rank Correlation Coefficient The Spearman correlation coefficient ( s ) is computed by using rank scores, which are defined in the section Scores on page This measure is appropriate only when both variables lie on an ordinal scale. The range of the Spearman correlation is 1 s 1. The Spearman correlation coefficient is estimated by r s D v = w and its asymptotic variance is var.r s / D 1 X X n 2 w 4 n ij.z ij Nz/ 2 i j

70 2276 Chapter 35: The FREQ Procedure where R1 i and C1 j are the row and column rank scores and v D X i X n ij R.i/C.j / j w D 1 p F G 12 X F D n 3 G D n 3 X i j n 3 i n 3 j R.i/ D R1 i n=2 C.j / D C1 j n=2 Nz D 1 X X n ij z ij n z ij D wv ij vw ij i j v ij D n R.i/C.j / C 1 X n il C.l/ C 1 X n kj R.k/ C 2 2 l k 1 X X X n kl R.k/ A w ij D n kl C.l/ C X l k>i k n F n 2 j 96w C Gn2 i See Snedecor and Cochran (1989) for more information. The variance under the null hypothesis that the correlation equals zero is computed as var 0.r s / D 1 X X n 2 w 2 n ij.v ij Nv/ 2 where Nv D X i X n ij v ij =n j i j Note that the asymptotic variance is derived for multinomial sampling in a contingency table framework, and it differs from the form obtained under the assumption that both variables are continuous and normally distributed. See Brown and Benedetti (1977) for details. PROC FREQ also provides an exact test for the Spearman correlation coefficient. You can request this test by specifying the SCORR option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. l>j

71 Statistical Computations 2277 Polychoric Correlation When you specify the PLCORR option in the TABLES statement, PROC FREQ computes the polychoric correlation. This measure of association is based on the assumption that the ordered, categorical variables of the frequency table have an underlying bivariate normal distribution. For 22 tables, the polychoric correlation is also known as the tetrachoric correlation. See Drasgow (1986) for an overview of polychoric correlation. The polychoric correlation coefficient is the maximum likelihood estimate of the product-moment correlation between the normal variables, estimating thresholds from the observed table frequencies. The range of the polychoric correlation is from 1 to 1. Olsson (1979) gives the likelihood equations and an asymptotic covariance matrix for the estimates. To estimate the polychoric correlation, PROC FREQ iteratively solves the likelihood equations by a Newton-Raphson algorithm that uses the Pearson correlation coefficient as the initial approximation. Iteration stops when the convergence measure falls below the convergence criterion or when the maximum number of iterations is reached, whichever occurs first. The CONVERGE= option sets the convergence criterion, and the default value is The MAXITER= option sets the maximum number of iterations, and the default value is 20. Lambda (Asymmetric) Asymmetric lambda,.c jr/, is interpreted as the probable improvement in predicting the column variable Y given knowledge of the row variable X. The range of asymmetric lambda is 0.C jr/ 1. Asymmetric lambda (C jr) is computed as P i.c jr/ D r i r n r and its asymptotic variance is where var..c jr// D n P i r X i.n r/ 3 r i C r i 2 X i.r i j l i D l/! r i D max j.n ij / r D max j.n j / c j D max.n ij / i c D max.n i / i The values of l i and l are determined as follows. Denote by l i the unique value of j such that r i D n ij, and let l be the unique value of j such that r D n j. Because of the uniqueness assumptions, ties in the frequencies or in the marginal totals must be broken in an arbitrary but consistent manner. In case of ties, l is defined as the smallest value of j such that r D n j.

72 2278 Chapter 35: The FREQ Procedure For those columns containing a cell.i; j / for which n ij D r i D c j, cs j records the row in which c j is assumed to occur. Initially cs j is set equal to 1 for all j. Beginning with i D 1, if there is at least one value j such that n ij D r i D c j, and if cs j D 1, then l i is defined to be the smallest such value of j, and cs j is set equal to i. Otherwise, if n il D r i, then l i is defined to be equal to l. If neither condition is true, then l i is taken to be the smallest value of j such that n ij D r i. The formulas for lambda asymmetric.rjc / can be obtained by interchanging the indices. See Goodman and Kruskal (1979) for more information. Lambda (Symmetric) The nondirectional lambda is the average of the two asymmetric lambdas,.c jr/ and.rjc /. Its range is 0 1. Lambda symmetric is computed as P i D r i C P j c j r c D w v 2n r c w and its asymptotic variance is computed as where var./ D 1 w 4 wvy 2w 2 n X X.n ij j j D l i ; i D k j / 2v 2.n n kl / i j r i D max j.n ij / r D max j.n j / c j D max.n ij / i c D max.n i / i w D 2n r c X v D 2n i r i X j c j x D X i.r i j l i D l/ C X j.c j j k j D k/ C r k C c l y D 8n w v 2x The definitions of l i and l are given in the previous section. The values k j and k are defined in a similar way for lambda asymmetric (RjC ). See Goodman and Kruskal (1979) for more information.

73 Statistical Computations 2279 Uncertainty Coefficients (Asymmetric) The uncertainty coefficient U.C jr/ measures the proportion of uncertainty (entropy) in the column variable Y that is explained by the row variable X. Its range is 0 U.C jr/ 1. The uncertainty coefficient is computed as U.C jr/ D.H.X/ C H.Y / H.XY // = H.Y / D v=w and its asymptotic variance is where var.u.c jr// D 1 X X n 2 w 4 i j n ij nij H.Y / ln C.H.X/ n i nj 2 H.XY // ln n v D H.X/ C H.Y / H.XY / w D H.Y / X H.X/ D H.Y / H.XY / D D i X j ni n nj n X X i j ln ni n nj ln n nij n nij ln n The formulas for the uncertainty coefficient U.RjC / can be obtained by interchanging the indices. See Theil (1972, pp ) and Goodman and Kruskal (1979) for more information. Uncertainty Coefficient (Symmetric) The uncertainty coefficient U is the symmetric version of the two asymmetric uncertainty coefficients. Its range is 0 U 1. The uncertainty coefficient is computed as U D 2.H.X/ C H.Y / H.XY // =.H.X/ C H.Y // and its asymptotic variance is var.u / D 4 X i X j ni n n ij H.XY / ln j n 2.H.X/ C H.Y // ln n ij n n 2.H.X/ C H.Y // 4 where H.X/, H.Y /, and H.XY / are defined in the previous section. See Goodman and Kruskal (1979) for more information. 2

74 2280 Chapter 35: The FREQ Procedure Binomial Proportion If you specify the BINOMIAL option in the TABLES statement, PROC FREQ computes the binomial proportion for one-way tables. By default, this is the proportion of observations in the first variable level that appears in the output. (You can use the LEVEL= option to specify a different level for the proportion.) The binomial proportion is computed as Op D n 1 = n where n 1 is the frequency of the first (or designated) level and n is the total frequency of the one-way table. The standard error of the binomial proportion is computed as se. Op/ D p Op.1 Op/ = n Confidence Limits By default, PROC FREQ provides asymptotic and exact (Clopper-Pearson) confidence limits for the binomial proportion. If you do not specify any confidence limit requests with binomial-options, PROC FREQ computes the standard Wald asymptotic confidence limits. You can also request Agresti-Coull, Jeffreys, and Wilson (score) confidence limits for the binomial proportion. See Brown, Cai, and DasGupta (2001), Agresti and Coull (1998), and Newcombe (1998) for details about these binomial confidence limits, including comparisons of their performance. Wald Confidence Limits The standard Wald asymptotic confidence limits are based on the normal approximation to the binomial distribution. PROC FREQ computes the Wald confidence limits for the binomial proportion as Op. z =2 se. Op/ / where z =2 is the =2/th percentile of the standard normal distribution. The confidence level is determined by the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits. If you specify the CORRECT binomial-option or the BINOMIALC option, PROC FREQ includes a continuity correction of 1=2n in the Wald asymptotic confidence limits. The purpose of this correction is to adjust for the difference between the normal approximation and the binomial distribution, which is a discrete distribution. See Fleiss, Levin, and Paik (2003) for more information. With the continuity correction, the asymptotic confidence limits for the binomial proportion are computed as Op. z =2 se. Op/ C.1=2n/ / Agresti-Coull Confidence Limits If you specify the AGRESTICOULL binomial-option, PROC FREQ computes Agresti-Coull confidence limits for the binomial proportion as where Qp. z =2 p Qp.1 Qp/ = Qn / Qn 1 D n 1 C.z =2 /=2 Qn D n C z 2 =2 Qp D Qn 1 = Qn

75 Statistical Computations 2281 The Agresti-Coull confidence interval has the same basic form as the standard Wald interval but uses Qp in place of Op. For = 0.05, the value of z =2 is close to 2, and this interval is the add 2 successes and 2 failures adjusted Wald interval in Agresti and Coull (1998). Jeffreys Confidence Limits If you specify the JEFFREYS binomial-option, PROC FREQ computes the Jeffreys confidence limits for the binomial proportion as ˇ. =2; n 1 C 1=2; n n 1 C 1=2/; ˇ.1 =2; n 1 C 1=2; n n 1 C 1=2/ where ˇ. ; b; c/ is the th percentile of the beta distribution with shape parameters b and c. The lower confidence limit is set to 0 when n 1 D 0, and the upper confidence limit is set to 1 when n 1 D n. This is an equal-tailed interval based on the noninformative Jeffreys prior for a binomial proportion. See Brown, Cai, and DasGupta (2001) for details. See Berger (1985) for information about using beta priors for inference on the binomial proportion. Wilson (Score) Confidence Limits If you specify the WILSON binomial-option, PROC FREQ computes Wilson confidence limits for the binomial proportion. These are also known as score confidence limits and are attributed to Wilson (1927). The confidence limits are based on inverting the normal test that uses the null proportion in the variance (the score test). Wilson confidence limits are the roots of jp Opj D z =2 p p.1 p/=n and are computed as Op C z 2 =2 =2n z =2r Op.1 Op/ C z 2 =2 =4n =n! = 1 C z 2 =2 =n The Wilson interval has been shown to have better performance than the Wald interval and the exact (Clopper-Pearson) interval. See Agresti and Coull (1998), Brown, Cai, and DasGupta (2001), and Newcombe (1998) for more information. Exact (Clopper-Pearson) Confidence Limits The exact or Clopper-Pearson confidence limits for the binomial proportion are constructed by inverting the equal-tailed test based on the binomial distribution. This method is attributed to Clopper and Pearson (1934). The exact confidence limits p L and p U satisfy the following equations, for n 1 D 1; 2; : : : n 1: nx xdn 1 n 1 X xd0! n x pl x p L/ n x D =2! n x pu x p U/ n x D =2 The lower confidence limit equals 0 when n 1 D 0, and the upper confidence limit equals 1 when n 1 D n.

76 2282 Chapter 35: The FREQ Procedure PROC FREQ computes the exact (Clopper-Pearson) confidence limits by using the F distribution as p L p U D D n n 1 C C n 1 F.1 =2; 2n 1 ; 2.n n 1 C 1/ / n n C.n 1 C 1/ F. =2; 2.n 1 C 1/; 2.n n 1 / / where F. ; b; c/ is the th percentile of the F distribution with b and c degrees of freedom. See Leemis and Trivedi (1996) for a derivation of this expression. Also see Collett (1991) for more information about exact binomial confidence limits. Because this is a discrete problem, the confidence coefficient (or coverage probability) of the exact (Clopper-Pearson) interval is not exactly (1 ) but is at least (1 ). Thus, this confidence interval is conservative. Unless the sample size is large, the actual coverage probability can be much larger than the target value. See Agresti and Coull (1998), Brown, Cai, and DasGupta (2001), and Leemis and Trivedi (1996) for more information about the performance of these confidence limits. Tests The BINOMIAL option provides an asymptotic equality test for the binomial proportion by default. You can also specify binomial-options to request tests of noninferiority, superiority, and equivalence for the binomial proportion. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ also computes exact p-values for the tests that you request with the binomial-options. Equality Test PROC FREQ computes an asymptotic test of the hypothesis that the binomial proportion equals p 0, where you can specify the value of p 0 with the P= binomial-option. If you do not specify a null value with P=, PROC FREQ uses p 0 D 0:5 by default. The binomial test statistic is computed as z D. Op p 0 /=se By default, the standard error is based on the null hypothesis proportion as se D p p 0.1 p 0 /=n If you specify the VAR=SAMPLE binomial-option, the standard error is computed from the sample proportion as se D p Op.1 Op/=n If you specify the CORRECT binomial-option or the BINOMIALC option, PROC FREQ includes a continuity correction in the asymptotic test statistic, towards adjusting for the difference between the normal approximation and the discrete binomial distribution. See Fleiss, Levin, and Paik (2003) for details. The continuity correction of.1=2n/ is subtracted from the numerator of the test statistic if. Op p 0 / is positive; otherwise, the continuity correction is added to the numerator. PROC FREQ computes one-sided and two-sided p-values for this test. When the test statistic z is greater than zero (its expected value under the null hypothesis), PROC FREQ computes the

77 Statistical Computations 2283 right-sided p-value, which is the probability of a larger value of the statistic occurring under the null hypothesis. A small right-sided p-value supports the alternative hypothesis that the true value of the proportion is greater than p 0. When the test statistic is less than or equal to zero, PROC FREQ computes the left-sided p-value, which is the probability of a smaller value of the statistic occurring under the null hypothesis. A small left-sided p-value supports the alternative hypothesis that the true value of the proportion is less than p 0. The one-sided p-value P 1 can be expressed as ( Prob.Z > z/ if z > 0 P 1 D Prob.Z < z/ if z 0 where Z has a standard normal distribution. The two-sided p-value is computed as P 2 D 2 P 1. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ also computes an exact test of the null hypothesis H 0 W p D p 0. To compute the exact test, PROC FREQ uses the binomial probability function,! Prob.X D x j p 0 / D n x p x 0.1 p 0/.n x/ for x D 0; 1; 2; : : : ; n where the variable X has a binomial distribution with parameters n and p 0. To compute the left-sided p-value, Prob.X n 1 /, PROC FREQ sums the binomial probabilities over x from zero to n 1. To compute the right-sided p-value, Prob.X n 1 /, PROC FREQ sums the binomial probabilities over x from n 1 to n. The exact one-sided p-value is the minimum of the left-sided and right-sided p-values, P 1 D min. Prob.X n 1 j p 0 /; Prob.X n 1 j p 0 / / and the exact two-sided p-value is computed as P 2 D 2 P 1. Noninferiority Test If you specify the NONINF binomial-option, PROC FREQ provides a noninferiority test for the binomial proportion. The null hypothesis for the noninferiority test is H 0 W p p 0 ı versus the alternative H a W p p 0 > ı where ı is the noninferiority margin and p 0 is the null proportion. Rejection of the null hypothesis indicates that the binomial proportion is not inferior to the null value. See Chow, Shao, and Wang (2003) for more information. You can specify the value of ı with the MARGIN= binomial-option, and you can specify p 0 with the P= binomial-option. By default, ı D 0:2 and p 0 D 0:5. PROC FREQ provides an asymptotic Wald test for noninferiority. The test statistic is computed as z D. Op p 0 / = se where p 0 is the noninferiority limit, p 0 D p 0 ı

78 2284 Chapter 35: The FREQ Procedure By default, the standard error is computed from the sample proportion as se D p Op.1 Op/=n If you specify the VAR=NULL binomial-option, the standard error is based on the noninferiority limit (determined by the null proportion and the margin) as q se D p0.1 p 0 /=n If you specify the CORRECT binomial-option or the BINOMIALC option, PROC FREQ includes a continuity correction in the asymptotic test statistic z. The continuity correction of.1=2n/ is subtracted from the numerator of the test statistic if. Op p0 / is positive; otherwise, the continuity correction is added to the numerator. The p-value for the noninferiority test is P z D Prob.Z > z/ where Z has a standard normal distribution. As part of the noninferiority analysis, PROC FREQ provides asymptotic Wald confidence limits for the binomial proportion. These confidence limits are computed as described in the section Wald Confidence Limits on page 2280 but use the same standard error (VAR=NULL or VAR=SAMPLE) as the noninferiority test statistic z. The confidence coefficient is /% (Schuirmann 1999). By default, if you do not specify the ALPHA= option, the noninferiority confidence limits are 90% confidence limits. You can compare the confidence limits to the noninferiority limit, p 0 D p 0 ı. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ provides an exact noninferiority test for the binomial proportion. The exact p-value is computed by using the binomial probability function with parameters p0 and n,! kdn X n P x D.p0 k / k.1 p0.n k/ / kdn 1 See Chow, Shao, Wang (2003, p. 116) for details. If you request exact binomial statistics, PROC FREQ also includes exact (Clopper-Pearson) confidence limits for the binomial proportion in the equivalence analysis display. See the section Exact (Clopper-Pearson) Confidence Limits on page 2281 for details. Superiority Test If you specify the SUP binomial-option, PROC FREQ provides a superiority test for the binomial proportion. The null hypothesis for the superiority test is H 0 W p p 0 ı versus the alternative H a W p p 0 > ı where ı is the superiority margin and p 0 is the null proportion. Rejection of the null hypothesis indicates that the binomial proportion is superior to the null value. You can specify the value of ı with

79 Statistical Computations 2285 the MARGIN= binomial-option, and you can specify the value of p 0 with the P= binomial-option. By default, ı D 0:2 and p 0 D 0:5. The superiority analysis is identical to the noninferiority analysis but uses a positive value of the margin ı in the null hypothesis. The superiority limit equals p 0 C ı. The superiority computations follow those in the section Noninferiority Test on page 2283 but replace ı with ı. See Chow, Shao, and Wang (2003) for more information. Equivalence Test If you specify the EQUIV binomial-option, PROC FREQ provides an equivalence test for the binomial proportion. The null hypothesis for the equivalence test is H 0 W p p 0 ı L or p p 0 ı U versus the alternative H a W ı L < p p 0 < ı U where ı L is the lower margin, ı U is the upper margin, and p 0 is the null proportion. Rejection of the null hypothesis indicates that the binomial proportion is equivalent to the null value. See Chow, Shao, and Wang (2003) for more information. You can specify the value of the margins ı L and ı U with the MARGIN= binomial-option. If you do not specify MARGIN=, PROC FREQ uses lower and upper margins of 0.2 and 0.2 by default. If you specify a single margin value ı, PROC FREQ uses lower and upper margins of ı and ı. You can specify the null proportion p 0 with the P= binomial-option. By default, p 0 D 0:5. PROC FREQ computes two one-sided tests (TOST) for equivalence analysis (Schuirmann 1987). The TOST approach includes a right-sided test for the lower margin and a left-sided test for the upper margin. The overall p-value is taken to be the larger of the two p-values from the lower and upper tests. For the lower margin, the asymptotic Wald test statistic is computed as z L D. Op p L / = se where the lower equivalence limit is p L D p 0 C ı L By default, the standard error is computed from the sample proportion as se D p Op.1 Op/=n If you specify the VAR=NULL binomial-option, the standard error is based on the lower equivalence limit (determined by the null proportion and the lower margin) as q se D pl.1 p L /=n If you specify the CORRECT binomial-option or the BINOMIALC option, PROC FREQ includes a continuity correction in the asymptotic test statistic z L. The continuity correction of.1=2n/ is

80 2286 Chapter 35: The FREQ Procedure subtracted from the numerator of the test statistic. Op the continuity correction is added to the numerator. pl / if the numerator is positive; otherwise, The p-value for the lower margin test is P z;l D Prob.Z > z L / The asymptotic test for the upper margin is computed similarly. The Wald test statistic is z U D. Op p U / = se where the upper equivalence limit is p U D p 0 C ı U By default, the standard error is computed from the sample proportion. If you specify the VAR=NULL binomial-option, the standard error is based on the upper equivalence limit as q se D pu.1 p U /=n If you specify the CORRECT binomial-option or the BINOMIALC option, PROC FREQ includes a continuity correction of.1=2n/ in the asymptotic test statistic z U. The p-value for the upper margin test is P z;u D Prob.Z < z U / Based on the two one-sided tests (TOST), the overall p-value for the test of equivalence equals the larger p-value from the lower and upper margin tests, which can be expressed as P z D max.p z;l ; P z;u / As part of the equivalence analysis, PROC FREQ provides asymptotic Wald confidence limits for the binomial proportion. These confidence limits are computed as described in the section Wald Confidence Limits on page 2280, but use the same standard error (VAR=NULL or VAR=SAMPLE) as the equivalence test statistics and have a confidence coefficient of /% (Schuirmann 1999). By default, if you do not specify the ALPHA= option, the equivalence confidence limits are 90% limits. If you specify VAR=NULL, separate standard errors are computed for the lower and upper margin tests, each based on the null proportion and the corresponding (lower or upper) margin. The confidence limits are computed by using the maximum of these two standard errors. You can compare the confidence limits to the equivalence limits,.p 0 C ı L ; p 0 C ı U /. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ also provides an exact equivalence test by using two one-sided exact tests (TOST). The procedure computes lower and upper margin exact tests by using the binomial probability function as described in the section Noninferiority Test on page The overall exact p-value for the equivalence test is taken to be the larger p-value from the lower and upper margin exact tests. If you request exact statistics, PROC FREQ also includes exact (Clopper-Pearson) confidence limits in the equivalence analysis display. The confidence coefficient is /% (Schuirmann 1999). See the section Exact (Clopper-Pearson) Confidence Limits on page 2281 for details.

81 Statistical Computations 2287 Risks and Risk Differences The RISKDIFF option in the TABLES statement provides estimates of risks (or binomial proportions) and risk differences for 2 2 tables. This analysis might be appropriate when comparing the proportion of some characteristic for two groups, where row 1 and row 2 correspond to the two groups, and the columns correspond to two possible characteristics or outcomes. For example, the row variable might be a treatment or dose, and the column variable might be the response. See Collett (1991), Fleiss, Levin, and Paik (2003), and Stokes, Davis, and Koch (2000) for more information. Let the frequencies of the 2 2 table be represented as follows. Column 1 Column 2 Total Row 1 n 11 n 12 n 1 Row 2 n 21 n 22 n 2 Total n 1 n 2 n For column 1 and column 2, PROC FREQ provides estimates of the row 1 risk (or proportion), the row 2 risk, the overall risk and the risk difference. The risk difference is defined as the row 1 risk minus the row 2 risk. The risks are binomial proportions of their rows (row 1, row 2, or overall), and the computation of their standard errors and confidence limits follow the binomial proportion computations, which are described in the section Binomial Proportion on page The column 1 risk for row 1 is the proportion of row 1 observations classified in column 1, p 1 D n 11 = n 1 This estimates the conditional probability of the column 1 response, given the first level of the row variable. The column 1 risk for row 2 is the proportion of row 2 observations classified in column 1, p 2 D n 21 = n 2 and the overall column 1 risk is the proportion of all observations classified in column 1, p D n 1 = n The column 1 risk difference compares the risks for the two rows, and it is computed as the column 1 risk for row 1 minus the column 1 risk for row 2, d D p 1 p 2 The risks and risk difference are defined similarly for column 2. The standard error of the column 1 risk for row i is computed as se.p i / D p p i.1 p i / = n 1 The standard error of the overall column 1 risk is computed as se.p/ D p p.1 p/ = n If the two rows represent independent binomial samples, the standard error for the column 1 risk difference is computed as se.d/ D p var.p 1 / C var.p 2 / The standard errors are computed in a similar manner for the column 2 risks and risk difference.

82 2288 Chapter 35: The FREQ Procedure Confidence Limits By default, the RISKDIFF option provides standard Wald asymptotic confidence limits for the risks (row 1, row 2, and overall) and the risk difference. The risks are equivalent to binomial proportions of their corresponding rows, and the computations follow the methods in the section Wald Confidence Limits on page The standard Wald asymptotic confidence limits are based on the normal approximation to the binomial distribution. PROC FREQ computes the Wald confidence limits for the risks and risk differences as est. z =2 se.est/ / where est is the estimate, z =2 is the =2/th percentile of the standard normal distribution, and se.est/ is the standard error of the estimate. The confidence level is determined from the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits. If you specify the CORRECT riskdiff-option or the RISKDIFFC option, PROC FREQ includes continuity corrections in the Wald asymptotic confidence limits for the risks and risk differences. The purpose of a continuity correction is to adjust for the difference between the normal approximation and the binomial distribution, which is discrete. See Fleiss, Levin, and Paik (2003) for more information. With the continuity correction, the asymptotic confidence limits are computed as est. z =2 se.est/ C cc / where cc is the continuity correction. For the row 1 risk, cc D.1=2n 1 /; for the row 2 risk, cc D.1=2n 2 /; for the overall risk, cc D.1=2n/; and for the risk difference, cc D..1=n 1 C 1=n 2 /=2/. The column 1 and column 2 risks use the same continuity corrections. PROC FREQ also computes exact (Clopper-Pearson) confidence limits for the column 1, column 2, and overall risks. These confidence limits are constructed by inverting the equal-tailed test based on the binomial distribution. PROC FREQ uses the F distribution to compute the Clopper-Pearson confidence limits. See the section Exact (Clopper-Pearson) Confidence Limits on page 2281 for details. PROC FREQ does not provide exact confidence limits for the risk difference by default. If you specify the RISKDIFF option in the EXACT statement, PROC FREQ provides exact unconditional confidence limits for the risk difference, which are described in the section Exact Unconditional Confidence Limits for the Risk Difference on page Note that the conditional exact approach, which is the basis for other exact tests provided by PROC FREQ such as Fisher s exact test, does not apply to the risk difference due to nuisance parameters. See Agresti (1992) for more information. Tests You can specify riskdiff-options to request tests of the risk (or proportion) difference. You can request tests of equality, noninferiority, superiority, and equivalence for the risk difference. The test of equality is a standard Wald asymptotic test, available with or without a continuity correction. For noninferiority, superiority, and equivalence tests of the risk difference, the following test methods are provided: Wald (with and without continuity correction), Hauck-Anderson, Farrington-Manning, and Newcombe score (with and without continuity correction). You can specify the test method with the METHOD= riskdiff-option. By default, PROC FREQ uses METHOD=WALD.

83 Statistical Computations 2289 Equality Test If you specify the EQUAL riskdiff-option, PROC FREQ computes a test of equality, or a test of the null hypothesis that the risk difference equals zero. For the column 1 (or 2) risk difference, this test can be expressed as H 0 W d D 0 versus the alternative H a W d 0, where d D p 1 p 2 denotes the column 1 (or 2) risk difference. PROC FREQ computes a standard Wald asymptotic test, and the test statistic is z D O d=se. O d/ By default, the standard error is computed from the sample proportions as se. O d/ D p Op 1.1 Op 1 /=n 1 C Op 2.1 Op 2 /=n 2 If you specify the VAR=NULL riskdiff-option, the standard error is based on the null hypothesis that the row 1 and row 2 risks are equal, se. O d/ D p Op.1 Op/.1=n 1 C 1=n 2 / where Op D n 1 =n estimates the overall column 1 risk. If you specify the CORRECT riskdiff-option or the RISKDIFFC option, PROC FREQ includes a continuity correction in the asymptotic test statistic. If O d > 0, the continuity correction is subtracted from O d in the numerator of the test statistic; otherwise, the continuity correction is added to the numerator. The value of the continuity correction is.1=n 1 C 1=n 2 /=2. PROC FREQ computes one-sided and two-sided p-values for this test. When the test statistic z is greater than 0, PROC FREQ displays the right-sided p-value, which is the probability of a larger value occurring under the null hypothesis. The one-sided p-value can be expressed as ( Prob.Z > z/ if z > 0 P 1 D Prob.Z < z/ if z 0 where Z has a standard normal distribution. The two-sided p-value is computed as P 2 D 2 P 1. Noninferiority Test If you specify the NONINF riskdiff-option, PROC FREQ provides a noninferiority test for the risk difference, or the difference between two proportions. The null hypothesis for the noninferiority test is H 0 W p 1 p 2 ı versus the alternative H a W p 1 p 2 > ı where ı is the noninferiority margin. Rejection of the null hypothesis indicates that the row 1 risk is not inferior to the row 2 risk. See Chow, Shao, and Wang (2003) for more information. You can specify the value of ı with the MARGIN= riskdiff-option. By default, ı D 0:2. You can specify the test method with the METHOD= riskdiff-option. The following methods are available for the risk difference noninferiority analysis: Wald (with and without continuity correction), Hauck- Anderson, Farrington-Manning, and Newcombe score (with and without continuity correction). The Wald, Hauck-Anderson, and Farrington-Manning methods provide tests and corresponding test-based

84 2290 Chapter 35: The FREQ Procedure confidence limits; the Newcombe score method provides only confidence limits. If you do not specify METHOD=, PROC FREQ uses the Wald test by default. The confidence coefficient for the test-based confidence limits is /% (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are 90% confidence limits. You can compare the confidence limits to the noninferiority limit, ı. The following sections describe the noninferiority analysis methods for the risk difference. Wald Test If you specify the METHOD=WALD riskdiff-option, PROC FREQ provides an asymptotic Wald test of noninferiority for the risk difference. This is also the default method. The Wald test statistic is computed as z D. O d C ı/ = se. O d/ where ( O d D Op 1 Op 2 ) estimates the risk difference and ı is the noninferiority margin. By default, the standard error for the Wald test is computed from the sample proportions as se. O d/ D p Op 1.1 Op 1 /=n 1 C Op 2.1 Op 2 /=n 2 If you specify the VAR=NULL riskdiff-option, the standard error is based on the null hypothesis that the risk difference equals ı (Dunnett and Gent 1977). The standard error is computed as where se. O d/ D p Qp.1 Qp/=n 2 C. Qp ı/.1 Qp C ı/=n 1 Qp D.n 11 C n 21 C ın 1 /=n If you specify the CORRECT riskdiff-option or the RISKDIFFC option, a continuity correction is included in the test statistic. The continuity correction is subtracted from the numerator of the test statistic if the numerator is greater than zero; otherwise, the continuity correction is added to the numerator. The value of the continuity correction is.1=n 1 C 1=n 2 /=2. The p-value for the Wald noninferiority test is P z D Prob.Z > z/, where Z has a standard normal distribution. Hauck-Anderson Test If you specify the METHOD=HA riskdiff-option, PROC FREQ provides the Hauck-Anderson test for noninferiority. The Hauck-Anderson test statistic is computed as z D. O d C ı cc/ = s. O d/ where O d D Op 1 Op 2 and the standard error is computed from the sample proportions as se. O d/ D p Op 1.1 Op 1 /=.n 1 1/ C Op 2.1 Op 2 /=.n 2 1/ The Hauck-Anderson continuity correction cc is computed as cc D 1 = 2 min.n 1 ; n 2 /

85 Statistical Computations 2291 The p-value for the Hauck-Anderson noninferiority test is P z D Prob.Z > z/, where Z has a standard normal distribution. See Hauck and Anderson (1986) and Schuirmann (1999) for more information. Farrington-Manning Test If you specify the METHOD=FM riskdiff-option, PROC FREQ provides the Farrington-Manning test of noninferiority for the risk difference. The Farrington-Manning test statistic is computed as z D. O d C ı/ = se. O d/ where O d D Op 1 Op 2 and se. O d/ D p Qp 1.1 Qp 1 /=n 1 C Qp 2.1 Qp 2 /=n 2 where Qp 1 and Qp 2 are the maximum likelihood estimators of p 1 and p 2 under the null hypothesis that the risk difference equals ı. The p-value for the Farrington-Manning noninferiority test is then P z D Prob.Z > z/, where Z has a standard normal distribution. From Farrington and Manning (1990), the solution to the maximum likelihood equation is where Qp 1 D 2u cos.w/ b=3a and Qp 2 D Qp 1 C ı w D. C cos 1.v=u 3 //=3 v D b 3 =.3a/ 3 q bc=6a 2 C d=2a u D sign.v/ b 2 =.3a/ 2 c=3a a D 1 C b D.1 C C Op 1 C Op 2 ı. C 2// c D ı 2 ı.2 Op 1 C C 1/ C Op 1 C Op 2 d D Op 1 ı.1 ı/ D n 2 =n 1 Newcombe Score Confidence Limits If you specify the METHOD=SCORE riskdiff-option, PROC FREQ provides the Newcombe hybrid score (or Wilson) confidence limits for the risk difference. The confidence coefficient for the confidence limits is /% (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are 90% confidence limits. You can compare the confidence limits to the noninferiority limit, ı. The Newcombe score confidence limits for the risk difference are constructed from the Wilson score confidence limits for each of the two individual proportions. The confidence limits for the individual proportions are used in the standard error terms of the Wald confidence limits for the proportion difference. See Newcombe (1998) and Barker et al. (2001) for more information. Wilson score confidence limits for p 1 and p 2 are the roots of jp i Op i j D z p pi.1 p i /=n i

86 2292 Chapter 35: The FREQ Procedure for i D 1; 2. The confidence limits are computed as Op i C z 2 =2n q i z Op i.1 Op i / C z 2 =4n i =ni = 1 C z 2 =n i See the section Wilson (Score) Confidence Limits on page 2281 for details. Denote the lower and upper Wilson score confidence limits for p 1 as L 1 and U 1, and denote the lower and upper confidence limits for p 2 as L 2 and U 2. The Newcombe score confidence limits for the proportion difference (d D p 1 p 2 ) are computed as d L D. Op 1 Op 2 / z p L1.1 L 1 /=n 1 C U 2.1 U 2 /=n 2 d U D. Op 1 Op 2 / C z p U1.1 U 1 /=n 1 C L 2.1 L 2 /=n 2 If you specify the CORRECT riskdiff-option, PROC FREQ provides continuity-corrected Newcombe score confidence limits. By including a continuity correction of 1=2n i, the Wilson score confidence limits for the individual proportions are the roots of jp i Op i j 1=2n i D z p pi.1 p i /=n i These confidence limits for the individual proportions are then used in the standard error terms of the Wald confidence limits for the proportion difference to compute d L and d U. Superiority Test If you specify the SUP riskdiff-option, PROC FREQ provides a superiority test for the risk difference. The null hypothesis is H 0 W W p 1 p 2 ı versus the alternative H a W p 1 p 2 > ı where ı is the superiority margin. Rejection of the null hypothesis indicates that the row 1 proportion is superior to the row 2 proportion. You can specify the value of ı with the MARGIN= riskdiff-option. By default, ı D 0:2. The superiority analysis is identical to the noninferiority analysis but uses a positive value of the margin ı in the null hypothesis. The superiority computations follow those in the section Noninferiority Test on page 2289 by replacing ı by ı. See Chow, Shao, and Wang (2003) for more information. Equivalence Tests If you specify the EQUIV riskdiff-option, PROC FREQ provides an equivalence test for the risk difference, or the difference between two proportions. The null hypothesis for the equivalence test is H 0 W p 1 p 2 ı L or p 1 p 2 ı U versus the alternative H a W ı L < p 1 p 2 < ı U

87 Statistical Computations 2293 where ı L is the lower margin and ı U is the upper margin. Rejection of the null hypothesis indicates that the two binomial proportions are equivalent. See Chow, Shao, and Wang (2003) for more information. You can specify the value of the margins ı L and ı U with the MARGIN= riskdiff-option. If you do not specify MARGIN=, PROC FREQ uses lower and upper margins of 0.2 and 0.2 by default. If you specify a single margin value ı, PROC FREQ uses lower and upper margins of ı and ı. You can specify the test method with the METHOD= riskdiff-option. The following methods are available for the risk difference equivalence analysis: Wald (with and without continuity correction), Hauck- Anderson, Farrington-Manning, and Newcombe s score (with and without continuity correction). The Wald, Hauck-Anderson, and Farrington-Manning methods provide tests and corresponding test-based confidence limits; the Newcombe score method provides only confidence limits. If you do not specify METHOD=, PROC FREQ uses the Wald test by default. PROC FREQ computes two one-sided tests (TOST) for equivalence analysis (Schuirmann 1987). The TOST approach includes a right-sided test for the lower margin ı L and a left-sided test for the upper margin ı U. The overall p-value is taken to be the larger of the two p-values from the lower and upper tests. The section Noninferiority Test on page 2289 gives details about the Wald, Hauck-Anderson, Farrington-Manning and Newcombe score methods for the risk difference. The lower margin equivalence test statistic takes the same form as the noninferiority test statistic but uses the lower margin value ı L in place of ı. The upper margin equivalence test statistic take the same form as the noninferiority test statistic but uses the upper margin value ı U in place of ı. The test-based confidence limits for the risk difference are computed according to the equivalence test method that you select. If you specify METHOD=WALD with VAR=NULL, or METHOD=FM, separate standard errors are computed for the lower and upper margin tests. In this case, the test-based confidence limits are computed by using the maximum of these two standard errors. The confidence limits have a confidence coefficient of /% (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are 90% confidence limits. You can compare the confidence limits to the equivalence limits,.ı L ; ı U /. Exact Unconditional Confidence Limits for the Risk Difference If you specify the RISKDIFF option in the EXACT statement, PROC FREQ provides exact unconditional confidence limits for the risk difference. Unconditional computations differ from the exact conditional approach that PROC FREQ uses for other exact statistics such as Fisher s exact test. (See the section Exact Statistics on page 2314 for more information.) Exact conditional inference does not apply to the risk difference due to nuisance parameters. See Agresti (1992) for details. The unconditional approach eliminates nuisance parameters by maximizing the p-value over all possible values of the nuisance parameters (Santner and Snell 1980). Denote the proportion difference by d D p 1 p 2. For a 2 2 table with row totals n 1 and n 2, the joint probability function can be expressed in terms of the table cell frequencies and the parameters d and p 2,!! f.n 11 ; n 21 I n 1 ; n 2 ; d; p 2 / D n 1 n 11.d Cp 2 / n 11.1 d p 2/ n 1 n 11 n 2 n 21 p n p 2 / n 2 n 21

88 2294 Chapter 35: The FREQ Procedure When constructing confidence limits for the proportion difference, the parameter of interest is d and p 2 is a nuisance parameter. Denote the observed value of the proportion difference by d 0 D Op 1 Op 2. The =2/% confidence limits for d are computed as where d L D sup.d W P U.d / > =2/ d U D inf.d W P L.d / > =2/ P U.d / D sup p 2 P L.d / D sup p 2 X f.n 11 ; n 21 I n 1 ; n 2 ; d ; p 2 / A;D.a/d 0 X f.n 11 ; n 21 I n 1 ; n 2 ; d ; p 2 / A;D.a/d 0 The set A includes all 2 2 tables with row sums equal to n 1 and n 2, and D.a/ denotes the value of the proportion difference (p 1 p 2 ) for table a in A. To compute P U.d /, the sum includes probabilities of those tables for which (D.a/ d 0 ), where d 0 is the observed value of the proportion difference. For a fixed value of d, P U.d / is taken to be the maximum sum over all possible values of p 2. See Santner and Snell (1980) and Agresti and Min (2001) for details. This method of eliminating the nuisance parameter is considered to be a conservative approach (Agresti and Min 2001). Additionally, the confidence limits are conservative for small samples because this is a discrete problem; the confidence coefficient is not exactly (1 ) but is at least (1 ). See Agresti (1992) for more information. Odds Ratio and Relative Risks for 2 x 2 Tables Odds Ratio (Case-Control Studies) The odds ratio is a useful measure of association for a variety of study designs. For a retrospective design called a case-control study, the odds ratio can be used to estimate the relative risk when the probability of positive response is small (Agresti 2002). In a case-control study, two independent samples are identified based on a binary (yes-no) response variable, and the conditional distribution of a binary explanatory variable is examined, within fixed levels of the response variable. See Stokes, Davis, and Koch (2000) and Agresti (2007). The odds of a positive response (column 1) in row 1 is n 11 =n 12. Similarly, the odds of a positive response in row 2 is n 21 =n 22. The odds ratio is formed as the ratio of the row 1 odds to the row 2 odds. The odds ratio for a 2 2 table is defined as OR D n 11=n 12 n 21 =n 22 D n 11 n 22 n 12 n 21 The odds ratio can be any nonnegative number. When the row and column variables are independent, the true value of the odds ratio equals 1. An odds ratio greater than 1 indicates that the odds of a

89 Statistical Computations 2295 positive response are higher in row 1 than in row 2. Values less than 1 indicate the odds of positive response are higher in row 2. The strength of association increases with the deviation from 1. The transformation G D.OR 1/=.OR C 1/ transforms the odds ratio to the range. 1; 1/ with G D 0 when OR D 1; G D 1 when OR D 0; and G approaches 1 as OR approaches infinity. G is the gamma statistic, which PROC FREQ computes when you specify the MEASURES option. The asymptotic /% confidence limits for the odds ratio are OR exp. z p v/; OR exp.z p v/ where v D var.ln OR/ D 1 n 11 C 1 n 12 C 1 n 21 C 1 n 22 and z is the =2/th percentile of the standard normal distribution. If any of the four cell frequencies are zero, the estimates are not computed. When you specify the OR option in the EXACT statement, PROC FREQ computes exact confidence limits for the odds ratio. Because this is a discrete problem, the confidence coefficient for the exact confidence interval is not exactly (1 ) but is at least (1 ). Thus, these confidence limits are conservative. See Agresti (1992) for more information. PROC FREQ computes exact confidence limits for the odds ratio by using an algorithm based on Thomas (1971). See also Gart (1971). The following two equations are solved iteratively to determine the lower and upper confidence limits, 1 and 2 : n 1 X idn 11 n 11 X id0! n 1 i! n 1 i n 1 n 1 n 2 n 2 i i!! i 1 = i 2 = n 1 X id0 n 1 X id0! n 1 i! n 1 i n 1 n 1 n 2 n 2 i i!! i 1 D =2 i 2 D =2 When the odds ratio equals zero, which occurs when either n 11 D 0 or n 22 D 0, PROC FREQ sets the lower exact confidence limit to zero and determines the upper limit with level. Similarly, when the odds ratio equals infinity, which occurs when either n 12 D 0 or n 21 D 0, PROC FREQ sets the upper exact confidence limit to infinity and determines the lower limit with level. Relative Risks (Cohort Studies) These measures of relative risk are useful in cohort (prospective) study designs, where two samples are identified based on the presence or absence of an explanatory factor. The two samples are observed in future time for the binary (yes-no) response variable under study. Relative risk measures are also useful in cross-sectional studies, where two variables are observed simultaneously. See Stokes, Davis, and Koch (2000) and Agresti (2007) for more information. The column 1 relative risk is the ratio of the column 1 risk for row 1 to row 2. The column 1 risk for row 1 is the proportion of the row 1 observations classified in column 1, p 1j1 D n 11 = n 1

90 2296 Chapter 35: The FREQ Procedure Similarly, the column 1 risk for row 2 is p 1j2 D n 21 = n 2 The column 1 relative risk is then computed as RR 1 D p 1j1 = p 1j2 A relative risk greater than 1 indicates that the probability of positive response is greater in row 1 than in row 2. Similarly, a relative risk less than 1 indicates that the probability of positive response is less in row 1 than in row 2. The strength of association increases with the deviation from 1. Asymptotic /% confidence limits for the column 1 relative risk are computed as RR 1 exp. z p v/; RR 1 exp.z p v/ where v D var.ln RR 1 / D.1 p 1j1 /=n 11 C.1 p1j2 /=n 21 and z is the =2/th percentile of the standard normal distribution. If either n 11 or n 21 is zero, the estimates are not computed. PROC FREQ computes the column 2 relative risks in the same way. Cochran-Armitage Test for Trend The TREND option in the TABLES statement provides the Cochran-Armitage test for trend, which tests for trend in binomial proportions across levels of a single factor or covariate. This test is appropriate for a two-way table where one variable has two levels and the other variable is ordinal. The two-level variable represents the response, and the other variable represents an explanatory variable with ordered levels. When the two-way has two columns and R rows, PROC FREQ tests for trend across the R levels of the row variable, and the binomial proportion is computed as the proportion of observations in the first column. When the table has two rows and C columns, PROC FREQ tests for trend across the C levels of the column variable, and the binomial proportion is computed as the proportion of observations in the first row. The trend test is based on the regression coefficient for the weighted linear regression of the binomial proportions on the scores of the explanatory variable levels. See Margolin (1988) and Agresti (2002) for details. If the table has two columns and R rows, the trend test statistic is computed as T D RX n i1.r i NR/ = id1 q p 1.1 p 1 / s 2 where R i is the score of row i, NR is the average row score, and s 2 D RX n i.r i NR/ 2 id1 The SCORES= option in the TABLES statement determines the type of row scores used in computing the trend test (and other score-based statistics). The default is SCORES=TABLE. See the section

91 Statistical Computations 2297 Scores on page 2265 for details. For character variables, the table scores for the row variable are the row numbers (for example, 1 for the first row, 2 for the second row, and so on). For numeric variables, the table score for each row is the numeric value of the row level. When you perform the trend test, the explanatory variable might be numeric (for example, dose of a test substance), and the variable values might be appropriate scores. If the explanatory variable has ordinal levels that are not numeric, you can assign meaningful scores to the variable levels. Sometimes equidistant scores, such as the table scores for a character variable, might be appropriate. For more information on choosing scores for the trend test, see Margolin (1988). The null hypothesis for the Cochran-Armitage test is no trend, which means that the binomial proportion p i1 D n i1 =n i is the same for all levels of the explanatory variable. Under the null hypothesis, the trend statistic has an asymptotic standard normal distribution. PROC FREQ computes one-sided and two-sided p-values for the trend test. When the test statistic is greater than its null hypothesis expected value of zero, PROC FREQ displays the right-sided p-value, which is the probability of a larger value of the statistic occurring under the null hypothesis. A small right-sided p-value supports the alternative hypothesis of increasing trend in proportions from row 1 to row R. When the test statistic is less than or equal to zero, PROC FREQ displays the left-sided p-value. A small left-sided p-value supports the alternative of decreasing trend. The one-sided p-value for the trend test is computed as ( Prob.Z > T / if T > 0 P 1 D Prob.Z < T / if T 0 where Z has a standard normal distribution. The two-sided p-value is computed as P 2 D Prob.jZj > jt j/ PROC FREQ also provides exact p-values for the Cochran-Armitage trend test. You can request the exact test by specifying the TREND option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Jonckheere-Terpstra Test The JT option in the TABLES statement provides the Jonckheere-Terpstra test, which is a nonparametric test for ordered differences among classes. It tests the null hypothesis that the distribution of the response variable does not differ among classes. It is designed to detect alternatives of ordered class differences, which can be expressed as 1 2 R (or 1 2 R ), with at least one of the inequalities being strict, where i denotes the effect of class i. For such ordered alternatives, the Jonckheere-Terpstra test can be preferable to tests of more general class difference alternatives, such as the Kruskal Wallis test (produced by the WILCOXON option in the NPAR1WAY procedure). See Pirie (1983) and Hollander and Wolfe (1999) for more information about the Jonckheere-Terpstra test. The Jonckheere-Terpstra test is appropriate for a two-way table in which an ordinal column variable represents the response. The row variable, which can be nominal or ordinal, represents the classification variable. The levels of the row variable should be ordered according to the ordering you want the test to detect. The order of variable levels is determined by the ORDER= option in the PROC

92 2298 Chapter 35: The FREQ Procedure FREQ statement. The default is ORDER=INTERNAL, which orders by unformatted values. If you specify ORDER=DATA, PROC FREQ orders values according to their order in the input data set. For more information about how to order variable levels, see the ORDER= option. The Jonckheere-Terpstra test statistic is computed by first forming R.R counts M i;i 0, where i < i 0, for pairs of rows in the contingency table, 1/=2 Mann-Whitney M i;i 0 D f number of times X i;j < X i 0 ;j 0; j D 1; : : : ; n i:i j 0 D 1; : : : ; n i 0 : g C 1 2 f number of times X i;j D X i 0 ;j 0; j D 1; : : : ; n i:i j 0 D 1; : : : ; n i 0 : g where X i;j is response j in row i. The Jonckheere-Terpstra test statistic is computed as J D X X 1i< i 0 R M i;i 0 This test rejects the null hypothesis of no difference among classes for large values of J. Asymptotic p-values for the Jonckheere-Terpstra test are obtained by using the normal approximation for the distribution of the standardized test statistic. The standardized test statistic is computed as J D.J E 0.J // = p var 0.J / where E 0.J / and var 0.J / are the expected value and variance of the test statistic under the null hypothesis,! X E 0.J / D n 2 =4 where i n 2 i var 0.J / D A=72 C B=.36n.n 1/.n 2// C C=.8n.n 1// A D n.n 1/.2n C 5/ B D C D X i X i X n i.n i 1/.2n i C 5/ i! 0 1 n i.n i 1/.n i X n j.n j 1/.n j 2/ A! 0 1 n i.n i X n j.n j 1/ A j j X n j.n j 1/.2n j C 5/ PROC FREQ computes one-sided and two-sided p-values for the Jonckheere-Terpstra test. When the standardized test statistic is greater than its null hypothesis expected value of zero, PROC FREQ displays the right-sided p-value, which is the probability of a larger value of the statistic occurring under the null hypothesis. A small right-sided p-value supports the alternative hypothesis of increasing order from row 1 to row R. When the standardized test statistic is less than or equal to zero, PROC FREQ displays the left-sided p-value. A small left-sided p-value supports the alternative of decreasing order from row 1 to row R. j

93 Statistical Computations 2299 The one-sided p-value for the Jonckheere-Terpstra test, P 1, is computed as ( Prob.Z > J / if J > 0 P 1 D Prob.Z < J / if J 0 where Z has a standard normal distribution. The two-sided p-value, P 2, is computed as P 2 D Prob.jZj > jj j/ PROC FREQ also provides exact p-values for the Jonckheere-Terpstra test. You can request the exact test by specifying the JT option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Tests and Measures of Agreement When you specify the AGREE option in the TABLES statement, PROC FREQ computes tests and measures of agreement for square tables (that is, for tables where the number of rows equals the number of columns). For two-way tables, these tests and measures include McNemar s test for 2 2 tables, Bowker s test of symmetry, the simple kappa coefficient, and the weighted kappa coefficient. For multiple strata (n-way tables, where n > 2), PROC FREQ also computes the overall simple kappa coefficient and the overall weighted kappa coefficient, as well as tests for equal kappas (simple and weighted) among strata. Cochran s Q is computed for multiway tables when each variable has two levels, that is, for h 2 2 tables. PROC FREQ computes the kappa coefficients (simple and weighted), their asymptotic standard errors, and their confidence limits when you specify the AGREE option in the TABLES statement. If you also specify the KAPPA option in the TEST statement, then PROC FREQ computes the asymptotic test of the hypothesis that simple kappa equals zero. Similarly, if you specify the WTKAP option in the TEST statement, PROC FREQ computes the asymptotic test for weighted kappa. In addition to the asymptotic tests described in this section, PROC FREQ provides exact p-values for McNemar s test, the simple kappa coefficient test, and the weighted kappa coefficient test. You can request these exact tests by specifying the corresponding options in the EXACT statement. See the section Exact Statistics on page 2314 for more information. The following sections provide the formulas that PROC FREQ uses to compute the AGREE statistics. For information about the use and interpretation of these statistics, see Agresti (2002), Agresti (2007), Fleiss, Levin, and Paik (2003), and the other references cited for each statistic. McNemar s Test PROC FREQ computes McNemar s test for 2 2 tables when you specify the AGREE option. McNemar s test is appropriate when you are analyzing data from matched pairs of subjects with a dichotomous (yes-no) response. It tests the null hypothesis of marginal homogeneity, or p 1 D p 1. McNemar s test is computed as Q M D.n 12 n 21 / 2 =.n 12 C n 21 /

94 2300 Chapter 35: The FREQ Procedure Under the null hypothesis, Q M has an asymptotic chi-square distribution with one degree of freedom. See McNemar (1947), as well as the general references cited in the preceding section. In addition to the asymptotic test, PROC FREQ also computes the exact p-value for McNemar s test when you specify the MCNEM option in the EXACT statement. Bowker s Test of Symmetry For Bowker s test of symmetry, the null hypothesis is that the cell proportions are symmetric, or that p ij D p j i for all pairs of table cells. For 2 2 tables, Bowker s test is identical to McNemar s test, and so PROC FREQ provides Bowker s test for square tables larger than 2 2. Bowker s test of symmetry is computed as XX Q B D.n ij n j i / 2 =.n ij C n j i / i<j For large samples, Q B has an asymptotic chi-square distribution with R.R under the null hypothesis of symmetry. See Bowker (1948) for details. 1/=2 degrees of freedom Simple Kappa Coefficient The simple kappa coefficient, introduced by Cohen (1960), is a measure of interrater agreement. PROC FREQ computes the simple kappa coefficient as O D.P o P e / =.1 P e / where P o D P i p ii and P e D P i p i:p :i. If the two response variables are viewed as two independent ratings of the n subjects, the kappa coefficient equals +1 when there is complete agreement of the raters. When the observed agreement exceeds chance agreement, kappa is positive, with its magnitude reflecting the strength of agreement. Although this is unusual in practice, kappa is negative when the observed agreement is less than chance agreement. The minimum value of kappa is between 1 and 0, depending on the marginal proportions. The asymptotic variance of the simple kappa coefficient is computed as where var.o/ D.A C B C / =.1 P e / 2 n A D X i p ii.1.p i C p i /.1 O// 2 XX B D.1 O/ 2 p ij.p i C p j / 2 i j C D. O P e.1 O/ / 2 See Fleiss, Cohen, and Everitt (1969) for details.

95 Statistical Computations 2301 PROC FREQ computes confidence limits for the simple kappa coefficient as O z =2 p var.o/ where z =2 is the =2/th percentile of the standard normal distribution. The value of is determined by the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits. To compute an asymptotic test for the kappa coefficient, PROC FREQ uses the standardized test statistic O, which has an asymptotic standard normal distribution under the null hypothesis that kappa equals zero. The standardized test statistic is computed as O D O = p var 0.O/ where var 0.O/ is the variance of the kappa coefficient under the null hypothesis,! X var 0.O/ D P e C Pe 2 p i p i.p i C p i / =.1 P e / 2 n See Fleiss, Levin, and Paik (2003) for details. i PROC FREQ also provides an exact test for the simple kappa coefficient. You can request the exact test by specifying the KAPPA or AGREE option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Weighted Kappa Coefficient The weighted kappa coefficient is a generalization of the simple kappa coefficient that uses weights to quantify the relative difference between categories. For 2 2 tables, the weighted kappa coefficient equals the simple kappa coefficient. PROC FREQ displays the weighted kappa coefficient only for tables larger than 2 2. PROC FREQ computes the kappa weights from the column scores, by using either Cicchetti-Allison weights or Fleiss-Cohen weights, both of which are described in the following section. The weights w ij are constructed so that 0 w ij < 1 for all i 6D j, w ii D 1 for all i, and w ij D w j i. The weighted kappa coefficient is computed as O w D P o.w/ P e.w/ = 1 Pe.w/ where P o.w/ D X i P e.w/ D X i X w ij p ij j X w ij p i p j j The asymptotic variance of the weighted kappa coefficient is 0 var.o w / X X p ij i j w ij.w i C w j /.1 O w / 2 O w P e.w/.1 1 O w / 2A =.1 P e.w/ / 2 n

96 2302 Chapter 35: The FREQ Procedure where w i D X j p j w ij w j D X i p i w ij See Fleiss, Cohen, and Everitt (1969) for details. PROC FREQ computes confidence limits for the weighted kappa coefficient as O w z =2 p var.o w / where z =2 is the =2/th percentile of the standard normal distribution. The value of is determined by the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits. To compute an asymptotic test for the weighted kappa coefficient, PROC FREQ uses the standardized test statistic O w, which has an asymptotic standard normal distribution under the null hypothesis that weighted kappa equals zero. The standardized test statistic is computed as O w D O w = p var 0.O w / where var 0.O w / is the variance of the weighted kappa coefficient under the null hypothesis, 0 var 0.O w / X X p i p j i j w ij.w i C w j / 2 1 P 2 A e.w/ =.1 P e.w/ / 2 n See Fleiss, Levin, and Paik (2003) for details. PROC FREQ also provides an exact test for the weighted kappa coefficient. You can request the exact test by specifying the WTKAPPA or AGREE option in the EXACT statement. See the section Exact Statistics on page 2314 for more information. Weights PROC FREQ computes kappa coefficient weights by using the column scores and one of the two available weight types. The column scores are determined by the SCORES= option in the TABLES statement. The two available types of kappa weights are Cicchetti-Allison and Fleiss-Cohen weights. By default, PROC FREQ uses Cicchetti-Allison weights. If you specify (WT=FC) with the AGREE option, then PROC FREQ uses Fleiss-Cohen weights to compute the weighted kappa coefficient. PROC FREQ computes Cicchetti-Allison kappa coefficient weights as w ij D 1 jc i C j j C C C 1 where C i is the score for column i and C is the number of categories or columns. See Cicchetti and Allison (1971) for details. The SCORES= option in the TABLES statement determines the type of column scores used to compute the kappa weights (and other score-based statistics). The default is SCORES=TABLE. See

97 Statistical Computations 2303 the section Scores on page 2265 for details. For numeric variables, table scores are the values of the variable levels. You can assign numeric values to the levels in a way that reflects their level of similarity. For example, suppose you have four levels and order them according to similarity. If you assign them values of 0, 2, 4, and 10, the Cicchetti-Allison kappa weights take the following values: w 12 = 0.8, w 13 = 0.6, w 14 = 0, w 23 = 0.8, w 24 = 0.2, and w 34 = 0.4. Note that when there are only two categories (that is, C = 2), the weighted kappa coefficient is identical to the simple kappa coefficient. If you specify (WT=FC) with the AGREE option in the TABLES statement, PROC FREQ computes Fleiss-Cohen kappa coefficient weights as w ij D 1.C i C j / 2.C C C 1 / 2 See Fleiss and Cohen (1973) for details. For the preceding example, the Fleiss-Cohen kappa weights are: w 12 = 0.96, w 13 = 0.84, w 14 = 0, w 23 = 0.96, w 24 = 0.36, and w 34 = Overall Kappa Coefficient When there are multiple strata, PROC FREQ combines the stratum-level estimates of kappa into an overall estimate of the supposed common value of kappa. Assume there are q strata, indexed by h D 1; 2; : : : ; q, and let var.o h / denote the variance of O h. The estimate of the overall kappa coefficient is computed as O T D qx hd1 O h var.o h / = qx hd1 1 var.o h / See Fleiss, Levin, and Paik (2003) for details. PROC FREQ computes an estimate of the overall weighted kappa in the same way. Tests for Equal Kappa Coefficients When there are multiple strata, the following chi-square statistic tests whether the stratum-level values of kappa are equal: Q K D qx.o h O T / 2 = var.o h / hd1 Under the null hypothesis of equal kappas for the q strata, Q K has an asymptotic chi-square distribution with q 1 degrees of freedom. See Fleiss, Levin, and Paik (2003) for more information. PROC FREQ computes a test for equal weighted kappa coefficients in the same way. Cochran s Q Test Cochran s Q is computed for multiway tables when each variable has two levels, that is, for tables. Cochran s Q statistic is used to test the homogeneity of the one-dimensional

98 2304 Chapter 35: The FREQ Procedure margins. Let m denote the number of variables and N denote the total number of subjects. Cochran s Q statistic is computed as 0 1! mx NX Q C D m.m T 2 A = mt j D1 j T 2 where T j is the number of positive responses for variable j, T is the total number of positive responses over all variables, and S k is the number of positive responses for subject k. Under the null hypothesis, Cochran s Q has an asymptotic chi-square distribution with m 1 degrees of freedom. See Cochran (1950) for details. When there are only two binary response variables (m D 2), Cochran s Q simplifies to McNemar s test. When there are more than two response categories, you can test for marginal homogeneity by using the repeated measures capabilities of the CATMOD procedure. kd1 S 2 k Tables with Zero Rows and Columns The AGREE statistics are defined only for square tables, where the number of rows equals the number of columns. If the table is not square, PROC FREQ does not compute AGREE statistics. In the kappa statistic framework, where two independent raters assign ratings to each of n subjects, suppose one of the raters does not use all possible r rating levels. If the corresponding table has r rows but only r 1 columns, then the table is not square and PROC FREQ does not compute AGREE statistics. To create a square table in this situation, use the ZEROS option in the WEIGHT statement, which requests that PROC FREQ include observations with zero weights in the analysis. Include zero-weight observations in the input data set to represent any rating levels that are not used by a rater, so that the input data set has at least one observation for each possible rater and rating combination. The analysis then includes all rating levels, even when all levels are not actually assigned by both raters. The resulting table (of rater 1 by rater 2) is a square table, and AGREE statistics can be computed. For more information, see the description of the ZEROS option. By default, PROC FREQ does not process observations that have zero weights, because these observations do not contribute to the total frequency count, and because any resulting zero-weight row or column causes many of the tests and measures of association to be undefined. However, kappa statistics are defined for tables with a zero-weight row or column, and the ZEROS option makes it possible to input zero-weight observations and construct the tables needed to compute kappas. Cochran-Mantel-Haenszel Statistics The CMH option in the TABLES statement gives a stratified statistical analysis of the relationship between the row and column variables after controlling for the strata variables in a multiway table. For example, for the table request A*B*C*D, the CMH option provides an analysis of the relationship between C and D, after controlling for A and B. The stratified analysis provides a way to adjust for the possible confounding effects of A and B without being forced to estimate parameters for them. The CMH analysis produces Cochran-Mantel-Haenszel statistics, which include the correlation statistic, the ANOVA (row mean scores) statistic, and the general association statistic. For 2 2 tables, the CMH option also provides Mantel-Haenszel and logit estimates of the common odds ratio and the common relative risks, as well as the Breslow-Day test for homogeneity of the odds ratios.

99 Statistical Computations 2305 Exact statistics are also available for stratified 2 2 tables. If you specify the EQOR option in the EXACT statement, PROC FREQ provides Zelen s exact test for equal odds ratios. If you specify the COMOR option in the EXACT statement, PROC FREQ provides exact confidence limits for the common odds ratio and an exact test that the common odds ratio equals one. Let the number of strata be denoted by q, indexing the strata by h D 1; 2; : : : ; q. Each stratum contains a contingency table with X representing the row variable and Y representing the column variable. For table h, denote the cell frequency in row i and column j by n hij, with corresponding row and column marginal totals denoted by n hi: and n h:j, and the overall stratum total by n h. Because the formulas for the Cochran-Mantel-Haenszel statistics are more easily defined in terms of matrices, the following notation is used. Vectors are presumed to be column vectors unless they are transposed. 0 /. n 0 hi D.n hi1 ; n hi2 ; : : : ; n hic /.1 C / n 0 D.n 0 h h1 ; n0 h2 ; : : : ; n0 /.1 RC / hr p hi D n hi = n h.1 1/ p hj D n hj = n h.1 1/ P 0 h D.p h1 ; p h2 ; : : : ; p hr /.1 R/ P 0 h D.p h1 ; p h2 ; : : : ; p hc /.1 C / Assume that the strata are independent and that the marginal totals of each stratum are fixed. The null hypothesis, H 0, is that there is no association between X and Y in any of the strata. The corresponding model is the multiple hypergeometric; this implies that, under H 0, the expected value and covariance matrix of the frequencies are, respectively, where m h D EŒn h j H 0 D n h.p h P h / varœn h j H 0 D c.d Ph P h P 0 h /.D Ph P h P 0 h / c D n 2 h =.n h 1/ and where denotes Kronecker product multiplication and D a is a diagonal matrix with the elements of a on the main diagonal. The generalized CMH statistic (Landis, Heyman, and Koch 1978) is defined as Q CMH D G 0 V G 1 G where G D X h B h.n h m h / V G D X h B h.var.n h j H 0 // B 0 h

100 2306 Chapter 35: The FREQ Procedure and where B h D C h R h is a matrix of fixed constants based on column scores C h and row scores R h. When the null hypothesis is true, the CMH statistic has an asymptotic chi-square distribution with degrees of freedom equal to the rank of B h. If V G is found to be singular, PROC FREQ prints a message and sets the value of the CMH statistic to missing. PROC FREQ computes three CMH statistics by using this formula for the generalized CMH statistic, with different row and column score definitions for each statistic. The CMH statistics that PROC FREQ computes are the correlation statistic, the ANOVA (row mean scores) statistic, and the general association statistic. These statistics test the null hypothesis of no association against different alternative hypotheses. The following sections describe the computation of these CMH statistics. CAUTION: The CMH statistics have low power for detecting an association in which the patterns of association for some of the strata are in the opposite direction of the patterns displayed by other strata. Thus, a nonsignificant CMH statistic suggests either that there is no association or that no pattern of association has enough strength or consistency to dominate any other pattern. Correlation Statistic The correlation statistic, popularized by Mantel and Haenszel (1959) and Mantel (1963), has one degree of freedom and is known as the Mantel-Haenszel statistic. The alternative hypothesis for the correlation statistic is that there is a linear association between X and Y in at least one stratum. If either X or Y does not lie on an ordinal (or interval) scale, then this statistic is not meaningful. To compute the correlation statistic, PROC FREQ uses the formula for the generalized CMH statistic with the row and column scores determined by the SCORES= option in the TABLES statement. See the section Scores on page 2265 for more information about the available score types. The matrix of row scores R h has dimension 1 R, and the matrix of column scores C h has dimension 1 C. When there is only one stratum, this CMH statistic reduces to.n 1/r 2, where r is the Pearson correlation coefficient between X and Y. When nonparametric (RANK or RIDIT) scores are specified, the statistic reduces to.n 1/rs 2, where r s is the Spearman rank correlation coefficient between X and Y. When there is more than one stratum, this CMH statistic becomes a stratum-adjusted correlation statistic. ANOVA (Row Mean Scores) Statistic The ANOVA statistic can be used only when the column variable Y lies on an ordinal (or interval) scale so that the mean score of Y is meaningful. For the ANOVA statistic, the mean score is computed for each row of the table, and the alternative hypothesis is that, for at least one stratum, the mean scores of the R rows are unequal. In other words, the statistic is sensitive to location differences among the R distributions of Y. The matrix of column scores C h has dimension 1 C, and the column scores are determined by the SCORES= option.

101 Statistical Computations 2307 The matrix of row scores R h has dimension.r R h D ŒI R 1 ; J R 1 1/ R and is created internally by PROC FREQ as where I R 1 is an identity matrix of rank R 1 and J R 1 is an.r 1/ 1 vector of ones. This matrix has the effect of forming R 1 independent contrasts of the R mean scores. When there is only one stratum, this CMH statistic is essentially an analysis of variance (ANOVA) statistic in the sense that it is a function of the variance ratio F statistic that would be obtained from a one-way ANOVA on the dependent variable Y. If nonparametric scores are specified in this case, then the ANOVA statistic is a Kruskal-Wallis test. If there is more than one stratum, then this CMH statistic corresponds to a stratum-adjusted ANOVA or Kruskal-Wallis test. In the special case where there is one subject per row and one subject per column in the contingency table of each stratum, this CMH statistic is identical to Friedman s chi-square. See Example 35.9 for an illustration. General Association Statistic The alternative hypothesis for the general association statistic is that, for at least one stratum, there is some kind of association between X and Y. This statistic is always interpretable because it does not require an ordinal scale for either X or Y. For the general association statistic, the matrix R h is the same as the one used for the ANOVA statistic. The matrix C h is defined similarly as C h D ŒI C 1 ; J C 1 PROC FREQ generates both score matrices internally. When there is only one stratum, then the general association CMH statistic reduces to Q P.n 1/=n, where Q P is the Pearson chi-square statistic. When there is more than one stratum, then the CMH statistic becomes a stratum-adjusted Pearson chi-square statistic. Note that a similar adjustment can be made by summing the Pearson chi-squares across the strata. However, the latter statistic requires a large sample size in each stratum to support the resulting chi-square distribution with q.r 1/.C 1/ degrees of freedom. The CMH statistic requires only a large overall sample size because it has only.r 1/.C 1/ degrees of freedom. See Cochran (1954); Mantel and Haenszel (1959); Mantel (1963); Birch (1965); and Landis, Heyman, and Koch (1978). Mantel-Fleiss Criterion If you specify the MF option in parentheses following the CMH option in the TABLES statement, PROC FREQ computes the Mantel-Fleiss criterion for stratified 2 2 tables. The Mantel-Fleiss criterion can be used to assess the validity of the chi-square approximation for the distribution of the Mantel-Haenszel statistic for 2 2 tables. See Mantel and Fleiss (1980), Mantel and Haenszel (1959), Stokes, Davis, and Koch (2000), and Dimitrienko et al. (2005) for details. The Mantel-Fleiss criterion is computed as " # X X MF D min.n h11 / L ; m h11 h h " #! X X.n h11 / U m h11 h h

102 2308 Chapter 35: The FREQ Procedure where m h11 is the expected value of n h11 under the hypothesis of no association between the row and column variables in table h,.n h11 / L is the minimum possible value of the table cell frequency, and.n h11 / U is the maximum possible value, m h11 D n h1 n h1 = n h.n h11 / L D max. 0; n h1 n h2 /.n h11 / U D min. n h1 ; n h1 / The Mantel-Fleiss guideline accepts the validity of the Mantel-Haenszel approximation when the value of the criterion is at least 5. When the criterion is less than 5, PROC FREQ displays a warning. Adjusted Odds Ratio and Relative Risk Estimates The CMH option provides adjusted odds ratio and relative risk estimates for stratified 2 2 tables. For each of these measures, PROC FREQ computes a Mantel-Haenszel estimate and a logit estimate. These estimates apply to n-way table requests in the TABLES statement, when the row and column variables both have two levels. For example, for the table request A*B*C*D, if the row and column variables C and D both have two levels, PROC FREQ provides odds ratio and relative risk estimates, adjusting for the confounding variables A and B. The choice of an appropriate measure depends on the study design. For case-control (retrospective) studies, the odds ratio is appropriate. For cohort (prospective) or cross-sectional studies, the relative risk is appropriate. See the section Odds Ratio and Relative Risks for 2 x 2 Tables on page 2294 for more information on these measures. Throughout this section, z denotes the =2/th percentile of the standard normal distribution. Odds Ratio, Case-Control Studies PROC FREQ provides Mantel-Haenszel and logit estimates for the common odds ratio for stratified 2 2 tables. The Mantel-Haenszel estimate of the common odds ratio is computed as!! X OR MH D n h11 n h22 =n h = n h12 n h21 =n h X h h It is always computed unless the denominator is zero. See Mantel and Haenszel (1959) and Agresti (2002) for details. To compute confidence limits for the common odds ratio, PROC FREQ uses the Greenland and Robins (1985) variance estimate for ln.or MH /. The =2/ confidence limits for the common odds ratio are OR MH exp. z O/; OR MH exp.z O/

103 Statistical Computations 2309 where O 2 D cvar. ln.or MH / / D P h.n h11 C n h22 /.n h11 n h22 /=n 2 h 2 P h n 2 h11 n h22 =n h C C P h Œ.n h11 C n h22 /.n h12 n h21 / C.n h12 C n h21 /.n h11 n h22 / =n 2 h 2 P h n P h11 n h22 =n h h n h12 n h21 =n h P h.n h12 C n h21 /.n h12 n h21 /=n 2 h 2 P h n 2 h12 n h21 =n h Note that the Mantel-Haenszel odds ratio estimator is less sensitive to small n h than the logit estimator. The adjusted logit estimate of the common odds ratio (Woolf 1955) is computed as OR L D exp X w h ln.or h / = X! w h h h and the corresponding /% confidence limits are s X s OR L z= A ; OR L w h h h where OR h is the odds ratio for stratum h, and w h D 1=var.ln.OR h // w h 1 1 A A If any table cell frequency in a stratum h is zero, PROC FREQ adds 0:5 to each cell of the stratum before computing OR h and w h (Haldane 1955) for the logit estimate. The procedure prints a warning when this occurs. Relative Risks, Cohort Studies PROC FREQ provides Mantel-Haenszel and logit estimates of the common relative risks for stratified 2 2 tables. The Mantel-Haenszel estimate of the common relative risk for column 1 is computed as!! X RR MH D n h11 n h2 = n h = n h21 n h1 = n h X h h It is always computed unless the denominator is zero. See Mantel and Haenszel (1959) and Agresti (2002) for more information.

104 2310 Chapter 35: The FREQ Procedure To compute confidence limits for the common relative risk, PROC FREQ uses the Greenland and Robins (1985) variance estimate for log.rr MH /. The =2/ confidence limits for the common relative risk are RR MH exp. z O/; RR MH exp.z O/ where O 2 D cvar. ln.rr MH / / D P h.n h1 n h2 n h1 n h11 n h21 n h /=n 2 h Ph n h11 n h2 =n h Ph n h21 n h1 =n h The adjusted logit estimate of the common relative risk for column 1 is computed as RR L D exp w h ln.rr h / = X! w h X h and the corresponding /% confidence limits are s X s RR L z = A ; RR L = w h h h where RR h is the column 1 relative risk estimate for stratum h and w h D 1 = var.ln.rr h // w h 1 1 A A If n h11 or n h21 is zero, then PROC FREQ adds 0:5 to each cell of the stratum before computing RR h and w h for the logit estimate. The procedure prints a warning when this occurs. See Kleinbaum, Kupper, and Morgenstern (1982, Sections 17.4 and 17.5) for details. Breslow-Day Test for Homogeneity of the Odds Ratios When you specify the CMH option, PROC FREQ computes the Breslow-Day test for stratified 2 2 tables. It tests the null hypothesis that the odds ratios for the q strata are equal. When the null hypothesis is true, the statistic has approximately a chi-square distribution with q 1 degrees of freedom. See Breslow and Day (1980) and Agresti (2007) for more information. The Breslow-Day statistic is computed as Q BD D X h.n h11 E.n h11 j OR MH // 2 = var.n h11 j OR MH / where E and var denote expected value and variance, respectively. The summation does not include any table with a zero row or column. If OR MH equals zero or if it is undefined, then PROC FREQ does not compute the statistic and prints a warning message. For the Breslow-Day test to be valid, the sample size should be relatively large in each stratum, and at least 80% of the expected cell counts should be greater than 5. Note that this is a stricter sample size requirement than the requirement for the Cochran-Mantel-Haenszel test for q 2 2 tables, in that each stratum sample size (not just the overall sample size) must be relatively large. Even when

105 Statistical Computations 2311 the Breslow-Day test is valid, it might not be very powerful against certain alternatives, as discussed in Breslow and Day (1980). If you specify the BDT option, PROC FREQ computes the Breslow-Day test with Tarone s adjustment, which subtracts an adjustment factor from Q BD to make the resulting statistic asymptotically chi-square. The Breslow-Day-Tarone statistic is computed as! 2 X Q BDT D Q BD.n h11 E.n h11 j OR MH // = X var.n h11 j OR MH / h See Tarone (1985), Jones et al. (1989), and Breslow (1996) for more information. h Zelen s Exact Test for Equal Odds Ratios If you specify the EQOR option in the EXACT statement, PROC FREQ computes Zelen s exact test for equal odds ratios for stratified 2 2 tables. Zelen s test is an exact counterpart to the Breslow-Day asymptotic test for equal odds ratios. The reference set for Zelen s test includes all possible q 2 2 tables with the same row, column, and stratum totals as the observed multiway table and with the same sum of cell.1; 1/ frequencies as the observed table. The test statistic is the probability of the observed q 2 2 table conditional on the fixed margins, which is a product of hypergeometric probabilities. The p-value for Zelen s test is the sum of all table probabilities that are less than or equal to the observed table probability, where the sum is computed over all tables in the reference set determined by the fixed margins and the observed sum of cell.1; 1/ frequencies. This test is similar to Fisher s exact test for two-way tables. See Zelen (1971), Hirji (2006), and Agresti (1992) for more information. PROC FREQ computes Zelen s exact test by using the polynomial multiplication algorithm of Hirji et al. (1996). Exact Confidence Limits for the Common Odds Ratio If you specify the COMOR option in the EXACT statement, PROC FREQ computes exact confidence limits for the common odds ratio for stratified 2 2 tables. This computation assumes that the odds ratio is constant over all the 2 2 tables. Exact confidence limits are constructed from the distribution of S D P h n h11, conditional on the marginal totals of the 2 2 tables. Because this is a discrete problem, the confidence coefficient for these exact confidence limits is not exactly (1 ) but is at least (1 ). Thus, these confidence limits are conservative. See Agresti (1992) for more information. PROC FREQ computes exact confidence limits for the common odds ratio by using an algorithm based on Vollset, Hirji, and Elashoff (1991). See also Mehta, Patel, and Gray (1985). Conditional on the marginal totals of 2 2 table h, let the random variable S h denote the frequency of table cell.1; 1/. Given the row totals n h1 and n h2 and column totals n h1 and n h2, the lower and upper bounds for S h are l h and u h, l h D max. 0; n h1 n h2 / u h D min. n h1 ; n h1 /

106 2312 Chapter 35: The FREQ Procedure Let C sh denote the hypergeometric coefficient,!! C sh D n h1 n h2 s h n h1 s h and let denote the common odds ratio. Then the conditional distribution of S h is P. S h D s h j n 1 ; n 1 ; n 2 / D C sh s h = x XD u h x D l h C x x Summing over all the 2 2 tables, S D P h S h, and the lower and upper bounds of S are l and u, l D X h l h and u D X h u h The conditional distribution of the sum S is P. S D s j n h1 ; n h1 ; n h2 I h D 1; : : : ; q / D C s s = xx D u x D l C x x where X C s D s 1 C::::Cs q D s Y h C sh! Let s 0 denote the observed sum of cell (1,1) frequencies over the q tables. The following two equations are solved iteratively for lower and upper confidence limits for the common odds ratio, 1 and 2 : xx D u x D s 0 xxd s 0 x D l x D u C x 1 x = X x D l x D u C x 2 x = X x D l C x x 1 D =2 C x x 2 D =2 When the observed sum s 0 equals the lower bound l, PROC FREQ sets the lower confidence limit to zero and determines the upper limit with level. Similarly, when the observed sum s 0 equals the upper bound u, PROC FREQ sets the upper confidence limit to infinity and determines the lower limit with level. When you specify the COMOR option in the EXACT statement, PROC FREQ also computes the exact test that the common odds ratio equals one. Setting D 1, the conditional distribution of the sum S under the null hypothesis becomes P 0. S D s j n h1 ; n h1 ; n h2 I h D 1; : : : ; q / D C s = xx D u x D l C x

107 Statistical Computations 2313 The point probability for this exact test is the probability of the observed sum s 0 under the null hypothesis, conditional on the marginals of the stratified 2 2 tables, and is denoted by P 0.s 0 /. The expected value of S under the null hypothesis is E 0.S/ D xx D u x D l x C x = xx D u x D l C x The one-sided exact p-value is computed from the conditional distribution as P 0.S >D s 0 / or P 0.S s 0 /, depending on whether the observed sum s 0 is greater or less than E 0.S/, xx D u P 1 D P 0. S >D s 0 / D C x = x D s 0 P 1 D P 0. S <D s 0 / D xxd s 0 x D l C x = xx D u x D l xx D u x D l C x C x if s 0 > E 0.S/ if s 0 E 0.S/ PROC FREQ computes two-sided p-values for this test according to three different definitions. A two-sided p-value is computed as twice the one-sided p-value, setting the result equal to one if it exceeds one, P a 2 D 2 P 1 Additionally, a two-sided p-value is computed as the sum of all probabilities less than or equal to the point probability of the observed sum s 0, summing over all possible values of s, l s u, P2 b D X P 0.s/ lsuwp 0.s/P 0.s 0 / Also, a two-sided p-value is computed as the sum of the one-sided p-value and the corresponding area in the opposite tail of the distribution, equidistant from the expected value, P c 2 D P 0. js E 0.S/j js 0 E 0.S/j / Gail-Simon Test for Qualitative Interactions The GAILSIMON option in the TABLES statement provides the Gail-Simon test for qualitative interaction for stratified 2 2 tables. See Gail and Simon (1985), Silvapulle (2001), and Dimitrienko et al. (2005) for details. The Gail-Simon test is based on the risk differences in stratified 22 tables, where the risk difference is defined as the row 1 risk (proportion in column 1) minus the row 2 risk. See the section Risks and Risk Differences on page 2287 for details. By default, the procedure uses column 1 risks to compute the Gail-Simon test. If you specify the COLUMN=2 option, the procedure uses column 2 risks.

108 2314 Chapter 35: The FREQ Procedure PROC FREQ computes the Gail-Simon test statistics as described in Gail and Simon (1985), Q D X h.d h =s h / 2 I.d h > 0/ QC D X h.d h =s h / 2 I.d h < 0/ Q D min.q ; QC/ where d h is the risk difference in table h, s h is the standard error of the risk difference, and I.d h > 0/ equals 1 if d h > 0 and 0 otherwise. Similarly, I.d h < 0/ equals 1 if d h < 0 and 0 otherwise. The q 2 2 tables (strata) are indexed by h D 1; 2; : : : ; q. The p-values for the Gail-Simon statistics are computed as p.q / D X h.1 F h.q // B.hI n D q; p D 0:5/ p.qc/ D X h.1 F h.qc// B.hI n D q; p D 0:5/ p.q/ D q 1 X hd1.1 F h.q// B.hI n D.q 1/; p D 0:5/ where F h./ is the cumulative chi-square distribution function with h degrees of freedom and B.hI n; p/ is the binomial probability function with parameters n and p. The statistic Q tests the null hypothesis of no qualitative interaction. The statistic Q tests the null hypothesis of positive risk differences. A small p-value for Q indicates negative differences; similarly, a small p-value for QC indicates positive risk differences. Exact Statistics Exact statistics can be useful in situations where the asymptotic assumptions are not met, and so the asymptotic p-values are not close approximations for the true p-values. Standard asymptotic methods involve the assumption that the test statistic follows a particular distribution when the sample size is sufficiently large. When the sample size is not large, asymptotic results might not be valid, with the asymptotic p-values differing perhaps substantially from the exact p-values. Asymptotic results might also be unreliable when the distribution of the data is sparse, skewed, or heavily tied. See Agresti (2007) and Bishop, Fienberg, and Holland (1975) for more information. Exact computations are based on the statistical theory of exact conditional inference for contingency tables, reviewed by Agresti (1992). In addition to computation of exact p-values, PROC FREQ provides the option of estimating exact p-values by Monte Carlo simulation. This can be useful for problems that are so large that exact computations require a great amount of time and memory, but for which asymptotic approximations might not be sufficient.

109 Statistical Computations 2315 Exact statistics are available for many PROC FREQ tests. For one-way tables, PROC FREQ provides exact p-values for the binomial proportion tests and the chi-square goodness-of-fit test. Exact confidence limits are available for the binomial proportion. For two-way tables, PROC FREQ provides exact p-values for the following tests: Pearson chi-square test, likelihood-ratio chi-square test, Mantel-Haenszel chi-square test, Fisher s exact test, Jonckheere-Terpstra test, and Cochran- Armitage test for trend. PROC FREQ also computes exact p-values for tests of the following statistics: Kendall s tau-b, Stuart s tau-c, Somers D.C jr/, Somers D.RjC /, Pearson correlation coefficient, Spearman correlation coefficient, simple kappa coefficient, and weighted kappa coefficient. For 2 2 tables, PROC FREQ provides exact confidence limits for the odds ratio, exact unconditional confidence limits for the proportion difference, and McNemar s exact test. For stratified 2 2 tables, PROC FREQ provides Zelen s exact test for equal odds ratios, exact confidence limits for the common odds ratio, and an exact test for the common odds ratio. The following sections summarize the exact computational algorithms, define the exact p-values that PROC FREQ computes, discuss the computational resource requirements, and describe the Monte Carlo estimation option. Computational Algorithms PROC FREQ computes exact p-values for general R C tables by using the network algorithm developed by Mehta and Patel (1983). This algorithm provides a substantial advantage over direct enumeration, which can be very time-consuming and feasible only for small problems. See Agresti (1992) for a review of algorithms for computation of exact p-values, and see Mehta, Patel, and Tsiatis (1984) and Mehta, Patel, and Senchaudhuri (1991) for information about the performance of the network algorithm. The reference set for a given contingency table is the set of all contingency tables with the observed marginal row and column sums. Corresponding to this reference set, the network algorithm forms a directed acyclic network consisting of nodes in a number of stages. A path through the network corresponds to a distinct table in the reference set. The distances between nodes are defined so that the total distance of a path through the network is the corresponding value of the test statistic. At each node, the algorithm computes the shortest and longest path distances for all the paths that pass through that node. For statistics that can be expressed as a linear combination of cell frequencies multiplied by increasing row and column scores, PROC FREQ computes shortest and longest path distances by using the algorithm of Agresti, Mehta, and Patel (1990). For statistics of other forms, PROC FREQ computes an upper bound for the longest path and a lower bound for the shortest path by following the approach of Valz and Thompson (1994). The longest and shortest path distances or bounds for a node are compared to the value of the test statistic to determine whether all paths through the node contribute to the p-value, none of the paths through the node contribute to the p-value, or neither of these situations occurs. If all paths through the node contribute, the p-value is incremented accordingly, and these paths are eliminated from further analysis. If no paths contribute, these paths are eliminated from the analysis. Otherwise, the algorithm continues, still processing this node and the associated paths. The algorithm finishes when all nodes have been accounted for. In applying the network algorithm, PROC FREQ uses full numerical precision to represent all statistics, row and column scores, and other quantities involved in the computations. Although it is possible to use rounding to improve the speed and memory requirements of the algorithm, PROC FREQ does not do this because it can result in reduced accuracy of the p-values.

110 2316 Chapter 35: The FREQ Procedure For one-way tables, PROC FREQ computes the exact chi-square goodness-of-fit test by the method of Radlow and Alf (1975). PROC FREQ generates all possible one-way tables with the observed total sample size and number of categories. For each possible table, PROC FREQ compares its chi-square value with the value for the observed table. If the table s chi-square value is greater than or equal to the observed chi-square, PROC FREQ increments the exact p-value by the probability of that table, which is calculated under the null hypothesis by using the multinomial frequency distribution. By default, the null hypothesis states that all categories have equal proportions. If you specify null hypothesis proportions or frequencies by using the TESTP= or TESTF= option in the TABLES statement, then PROC FREQ calculates the exact chi-square test based on that null hypothesis. Other exact computations are described in sections about the individual statistics. See the section Binomial Proportion on page 2280 for details about how PROC FREQ computes exact confidence limits and tests for the binomial proportion. See the section Odds Ratio and Relative Risks for 2 x 2 Tables on page 2294 for information about computation of exact confidence limits for the odds ratio for 2 2 tables. Also, see the sections Exact Unconditional Confidence Limits for the Risk Difference on page 2293, Exact Confidence Limits for the Common Odds Ratio on page 2311, and Zelen s Exact Test for Equal Odds Ratios on page Definition of p-values For several tests in PROC FREQ, the test statistic is nonnegative, and large values of the test statistic indicate a departure from the null hypothesis. Such nondirectional tests include the Pearson chisquare, the likelihood-ratio chi-square, the Mantel-Haenszel chi-square, Fisher s exact test for tables larger than 2 2, McNemar s test, and the one-way chi-square goodness-of-fit test. The exact p-value for a nondirectional test is the sum of probabilities for those tables having a test statistic greater than or equal to the value of the observed test statistic. There are other tests where it might be appropriate to test against either a one-sided or a two-sided alternative hypothesis. For example, when you test the null hypothesis that the true parameter value equals 0 (T D 0), the alternative of interest might be one-sided (T 0, or T 0) or two-sided (T 0). Such tests include the Pearson correlation coefficient, Spearman correlation coefficient, Jonckheere-Terpstra test, Cochran-Armitage test for trend, simple kappa coefficient, and weighted kappa coefficient. For these tests, PROC FREQ displays the right-sided p-value when the observed value of the test statistic is greater than its expected value. The right-sided p-value is the sum of probabilities for those tables for which the test statistic is greater than or equal to the observed test statistic. Otherwise, when the observed test statistic is less than or equal to the expected value, PROC FREQ displays the left-sided p-value. The left-sided p-value is the sum of probabilities for those tables for which the test statistic is less than or equal to the one observed. The one-sided p-value P 1 can be expressed as ( Prob. Test Statistic t / if t > E 0.T / P 1 D Prob. Test Statistic t / if t E 0.T / where t is the observed value of the test statistic and E 0.T / is the expected value of the test statistic under the null hypothesis. PROC FREQ computes the two-sided p-value as the sum of the one-sided p-value and the corresponding area in the opposite tail of the distribution of the statistic, equidistant from the expected value. The two-sided p-value P 2 can be expressed as P 2 D Prob. jtest Statistic E 0.T /j jt E 0.T /j/

111 Statistical Computations 2317 If you specify the POINT option in the EXACT statement, PROC FREQ also displays exact point probabilities for the test statistics. The exact point probability is the exact probability that the test statistic equals the observed value. Computational Resources PROC FREQ uses relatively fast and efficient algorithms for exact computations. These recently developed algorithms, together with improvements in computer power, now make it feasible to perform exact computations for data sets where previously only asymptotic methods could be applied. Nevertheless, there are still large problems that might require a prohibitive amount of time and memory for exact computations, depending on the speed and memory available on your computer. For large problems, consider whether exact methods are really needed or whether asymptotic methods might give results quite close to the exact results, while requiring much less computer time and memory. When asymptotic methods might not be sufficient for such large problems, consider using Monte Carlo estimation of exact p-values, as described in the section Monte Carlo Estimation on page A formula does not exist that can predict in advance how much time and memory are needed to compute an exact p-value for a certain problem. The time and memory required depend on several factors, including which test is being performed, the total sample size, the number of rows and columns, and the specific arrangement of the observations into table cells. Generally, larger problems (in terms of total sample size, number of rows, and number of columns) tend to require more time and memory. Additionally, for a fixed total sample size, time and memory requirements tend to increase as the number of rows and columns increases, because this corresponds to an increase in the number of tables in the reference set. Also for a fixed sample size, time and memory requirements increase as the marginal row and column totals become more homogeneous. See Agresti, Mehta, and Patel (1990) and Gail and Mantel (1977) for more information. At any time while PROC FREQ is computing exact p-values, you can terminate the computations by pressing the system interrupt key sequence (see the SAS Companion for your system) and choosing to stop computations. After you terminate exact computations, PROC FREQ completes all other remaining tasks. The procedure produces the requested output and reports missing values for any exact p-values that were not computed by the time of termination. You can also use the MAXTIME= option in the EXACT statement to limit the amount of time PROC FREQ uses for exact computations. You specify a MAXTIME= value that is the maximum amount of clock time (in seconds) that PROC FREQ can use to compute an exact p-value. If PROC FREQ does not finish computing an exact p-value within that time, it terminates the computation and completes all other remaining tasks. Monte Carlo Estimation If you specify the option MC in the EXACT statement, PROC FREQ computes Monte Carlo estimates of the exact p-values instead of directly computing the exact p-values. Monte Carlo estimation can be useful for large problems that require a great amount of time and memory for exact computations but for which asymptotic approximations might not be sufficient. To describe the precision of each Monte Carlo estimate, PROC FREQ provides the asymptotic standard error and /% confidence limits. The confidence level is determined by the ALPHA= option in the EXACT statement,

112 2318 Chapter 35: The FREQ Procedure which, by default, equals 0.01 and produces 99% confidence limits. The N=n option in the EXACT statement specifies the number of samples that PROC FREQ uses for Monte Carlo estimation; the default is samples. You can specify a larger value for n to improve the precision of the Monte Carlo estimates. Because larger values of n generate more samples, the computation time increases. Alternatively, you can specify a smaller value of n to reduce the computation time. To compute a Monte Carlo estimate of an exact p-value, PROC FREQ generates a random sample of tables with the same total sample size, row totals, and column totals as the observed table. PROC FREQ uses the algorithm of Agresti, Wackerly, and Boyett (1979), which generates tables in proportion to their hypergeometric probabilities conditional on the marginal frequencies. For each sample table, PROC FREQ computes the value of the test statistic and compares it to the value for the observed table. When estimating a right-sided p-value, PROC FREQ counts all sample tables for which the test statistic is greater than or equal to the observed test statistic. Then the p-value estimate equals the number of these tables divided by the total number of tables sampled. O P MC D M = N M D number of samples with.test Statistic t/ N D total number of samples t D observed Test Statistic PROC FREQ computes left-sided and two-sided p-value estimates in a similar manner. For left-sided p-values, PROC FREQ evaluates whether the test statistic for each sampled table is less than or equal to the observed test statistic. For two-sided p-values, PROC FREQ examines the sample test statistics according to the expression for P 2 given in the section Definition of p-values on page The variable M is a binomially distributed variable with N trials and success probability p. It follows that the asymptotic standard error of the Monte Carlo estimate is q se. PO MC / D PO MC.1 O P MC / =.N 1/ PROC FREQ constructs asymptotic confidence limits for the p-values according to PO MC z =2 se. PO MC / where z =2 is the =2/th percentile of the standard normal distribution and the confidence level is determined by the ALPHA= option in the EXACT statement. When the Monte Carlo estimate O P MC equals 0, PROC FREQ computes the confidence limits for the p-value as. 0; 1.1=N / / When the Monte Carlo estimate O P MC equals 1, PROC FREQ computes the confidence limits as..1=n / ; 1 /

113 Computational Resources 2319 Computational Resources For each variable in a table request, PROC FREQ stores all of the levels in memory. If all variables are numeric and not formatted, this requires about 84 bytes for each variable level. When there are character variables or formatted numeric variables, the memory that is required depends on the formatted variable lengths, with longer formatted lengths requiring more memory. The number of levels for each variable is limited only by the largest integer that your operating environment can store. For any single crosstabulation table requested, PROC FREQ builds the entire table in memory, regardless of whether the table has zero cell counts. Thus, if the numeric variables A, B, and C each have 10 levels, PROC FREQ requires 2520 bytes to store the variable levels for the table request A*B*C, as follows: 3 variables * 10 levels/variable * 84 bytes/level In addition, PROC FREQ requires 8000 bytes to store the table cell frequencies 1000 cells * 8 bytes/cell even though there might be only 10 observations. When the variables have many levels or when there are many multiway tables, your computer might not have enough memory to construct the tables. If PROC FREQ runs out of memory while constructing tables, it stops collecting levels for the variable with the most levels and returns the memory that is used by that variable. The procedure then builds the tables that do not contain the disabled variables. If there is not enough memory for your table request and if increasing the available memory is impractical, you can reduce the number of multiway tables or variable levels. If you are not using the CMH or AGREE option in the TABLES statement to compute statistics across strata, reduce the number of multiway tables by using PROC SORT to sort the data set by one or more of the variables or by using the DATA step to create an index for the variables. Then remove the sorted or indexed variables from the TABLES statement and include a BY statement that uses these variables. You can also reduce memory requirements by using a FORMAT statement in the PROC FREQ step to reduce the number of levels. Additionally, reducing the formatted variable lengths reduces the amount of memory that is needed to store the variable levels. For more information about using formats, see the section Grouping with Formats on page 2260.

114 2320 Chapter 35: The FREQ Procedure Output Data Sets PROC FREQ produces two types of output data sets that you can use with other statistical and reporting procedures. You can request these data sets as follows: Specify the OUT= option in a TABLES statement. This creates an output data set that contains frequency or crosstabulation table counts and percentages Specify an OUTPUT statement. This creates an output data set that contains statistics. PROC FREQ does not display the output data sets. Use PROC PRINT, PROC REPORT, or any other SAS reporting tool to display an output data set. In addition to these two output data sets, you can create a SAS data set from any piece of PROC FREQ output by using the Output Delivery System. See the section ODS Table Names on page 2331 for more information. Contents of the TABLES Statement Output Data Set The OUT= option in the TABLES statement creates an output data set that contains one observation for each combination of variable values (or table cell) in the last table request. By default, each observation contains the frequency and percentage for the table cell. When the input data set contains missing values, the output data set also contains an observation with the frequency of missing values. The output data set includes the following variables: BY variables table request variables, such as A, B, C, and D in the table request A*B*C*D COUNT, which contains the table cell frequency PERCENT, which contains the table cell percentage If you specify the OUTEXPECT option in the TABLES statement for a two-way or multiway table, the output data set also includes expected frequencies. If you specify the OUTPCT option for a two-way or multiway table, the output data set also includes row, column, and table percentages. The additional variables are as follows: EXPECTED, which contains the expected frequency PCT_TABL, which contains the percentage of two-way table frequency, for n-way tables where n > 2 PCT_ROW, which contains the percentage of row frequency PCT_COL, which contains the percentage of column frequency

115 Output Data Sets 2321 If you specify the OUTCUM option in the TABLES statement for a one-way table, the output data set also includes cumulative frequencies and cumulative percentages. The additional variables are as follows: CUM_FREQ, which contains the cumulative frequency CUM_PCT, which contains the cumulative percentage The OUTCUM option has no effect for two-way or multiway tables. The following PROC FREQ statements create an output data set of frequencies and percentages: proc freq; tables A A*B / out=d; run; The output data set D contains frequencies and percentages for the table of A by B, which is the last table request listed in the TABLES statement. If A has two levels (1 and 2), B has three levels (1,2, and 3), and no table cell count is zero or missing, then the output data set D includes six observations, one for each combination of A and B levels. The first observation corresponds to A=1 and B=1; the second observation corresponds to A=1 and B=2; and so on. The data set includes the variables COUNT and PERCENT. The value of COUNT is the number of observations with the given combination of A and B levels. The value of PERCENT is the percentage of the total number of observations with that A and B combination. When PROC FREQ combines different variable values into the same formatted level, the output data set contains the smallest internal value for the formatted level. For example, suppose a variable X has the values 1.1., 1.4, 1.7, 2.1, and 2.3. When you submit the statement format X 1.; in a PROC FREQ step, the formatted levels listed in the frequency table for X are 1 and 2. If you create an output data set with the frequency counts, the internal values of the levels of X are 1.1 and 1.7. To report the internal values of X when you display the output data set, use a format of 3.1 for X. Contents of the OUTPUT Statement Output Data Set The OUTPUT statement creates a SAS data set that contains the statistics that PROC FREQ computes for the last table request. You specify which statistics to store in the output data set. There is an observation with the specified statistics for each stratum or two-way table. If PROC FREQ computes summary statistics for a stratified table, the output data set also contains a summary observation with those statistics. The OUTPUT data set can include the following variables. BY variables variables that identify the stratum, such as A and B in the table request A*B*C*D variables that contain the specified statistics

116 2322 Chapter 35: The FREQ Procedure The output data set also includes variables with the p-values and degrees of freedom, asymptotic standard error (ASE), or confidence limits when PROC FREQ computes these values for a specified statistic. The variable names for the specified statistics in the output data set are the names of the options enclosed in underscores. PROC FREQ forms variable names for the corresponding p-values, degrees of freedom, or confidence limits by combining the name of the option with the appropriate prefix from the following list: DF_ E_ L_ U_ E0_ Z_ degrees of freedom asymptotic standard error (ASE) lower confidence limit upper confidence limit ASE under the null hypothesis standardized value P_ P2_ PL_ PR_ XP_ XP2_ XPL_ XPR_ XPT_ XL_ XU_ p-value two-sided p-value left-sided p-value right-sided p-value exact p-value exact two-sided p-value exact left-sided p-value exact right-sided p-value exact point probability exact lower confidence limit exact upper confidence limit For example, variable names created for the Pearson chi-square, its degrees of freedom, and its p-values are _PCHI_, DF_PCHI, and P_PCHI, respectively. If the length of the prefix plus the statistic option exceeds eight characters, PROC FREQ truncates the option so that the name of the new variable is eight characters long. Displayed Output Number of Variable Levels Table If you specify the NLEVELS option in the PROC FREQ statement, PROC FREQ displays the Number of Variable Levels table. This table provides the number of levels for all variables named in the TABLES statements. PROC FREQ determines the variable levels from the formatted variable values. See Grouping with Formats on page 2260 for details. The Number of Variable Levels table contains the following information: Variable name Levels, which is the total number of levels of the variable

117 Displayed Output 2323 Number of Nonmissing Levels, if there are missing levels for any of the variables Number of Missing Levels, if there are missing levels for any of the variables One-Way Frequency Tables PROC FREQ displays one-way frequency tables for all one-way table requests in the TABLES statements, unless you specify the NOPRINT option in the PROC statement or the NOPRINT option in the TABLES statement. For a one-way table showing the frequency distribution of a single variable, PROC FREQ displays the name of the variable and its values. For each variable value or level, PROC FREQ displays the following information: Frequency count, which is the number of observations in the level Test Frequency count, if you specify the CHISQ and TESTF= options to request a chi-square goodness-of-fit test for specified frequencies Percent, which is the percentage of the total number of observations. (The NOPERCENT option suppresses this information.) Test Percent, if you specify the CHISQ and TESTP= options to request a chi-square goodnessof-fit test for specified percents. (The NOPERCENT option suppresses this information.) Cumulative Frequency count, which is the sum of the frequency counts for that level and all other levels listed above it in the table. The last cumulative frequency is the total number of nonmissing observations. (The NOCUM option suppresses this information.) Cumulative Percent, which is the percentage of the total number of observations in that level and in all other levels listed above it in the table. (The NOCUM or the NOPERCENT option suppresses this information.) The one-way table also displays the Frequency Missing, which is the number of observations with missing values. Statistics for One-Way Frequency Tables For one-way tables, two statistical options are available in the TABLES statement. The CHISQ option provides a chi-square goodness-of-fit test, and the BINOMIAL option provides binomial proportion statistics and tests. PROC FREQ displays the following information, unless you specify the NOPRINT option in the PROC statement: If you specify the CHISQ option for a one-way table, PROC FREQ provides a chi-square goodness-of-fit test, displaying the Chi-Square statistic, the degrees of freedom (DF), and the probability value (Pr > ChiSq). If you specify the CHISQ option in the EXACT statement, PROC FREQ also displays the exact probability value for this test. If you specify the POINT option with the CHISQ option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic.

118 2324 Chapter 35: The FREQ Procedure If you specify the BINOMIAL option for a one-way table, PROC FREQ displays the estimate of the binomial Proportion, which is the proportion of observations in the first class listed in the one-way table. PROC FREQ also displays the asymptotic standard error (ASE) and the asymptotic (Wald) and exact (Clopper-Pearson) confidence limits by default. For the binomial proportion test, PROC FREQ displays the asymptotic standard error under the null hypothesis (ASE Under H0), the standardized test statistic (Z), and the one-sided and two-sided probability values. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ also displays the exact one-sided and two-sided probability values for this test. If you specify the POINT option with the BINOMIAL option in the EXACT statement, PROC FREQ displays the exact point probability for the test. If you request additional binomial confidence limits by specifying binomial-options, PROC FREQ provides a table that displays the lower and upper confidence limits for each type that you request. In addition to the Wald and exact (Clopper-Pearson) confidence limits, you can request Agresti-Coull, Jeffreys, and Wilson (score) confidence limits for the binomial proportion. If you request a binomial noninferiority or superiority test by specifying the NONINF or SUP binomial-option, PROC FREQ displays the following information: the binomial Proportion, the test ASE (under H0 or Sample), the test statistic Z, the probability value, the noninferiority or superiority limit, and the test confidence limits. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ also provides the exact probability value for the test, and exact test confidence limits. If you request a binomial equivalence test by specifying the EQUIV binomial-option, PROC FREQ displays the binomial Proportion and the test ASE (under H0 or Sample). PROC FREQ displays two one-sided tests (TOST) for equivalence, which include test statistics (Z) and probability values for the Lower and Upper tests, together with the Overall probability value. PROC FREQ also displays the equivalence limits and the test-based confidence limits. If you specify the BINOMIAL option in the EXACT statement, PROC FREQ provides exact probability values for the TOST and exact test-based confidence limits. Multiway Tables PROC FREQ displays all multiway table requests in the TABLES statements, unless you specify the NOPRINT option in the PROC statement or the NOPRINT option in the TABLES statement. For two-way to multiway crosstabulation tables, the values of the last variable in the table request form the table columns. The values of the next-to-last variable form the rows. Each level (or combination of levels) of the other variables forms one stratum. There are three ways to display multiway tables in PROC FREQ. By default, PROC FREQ displays multiway tables as separate two-way crosstabulation tables for each stratum of the multiway table. Also by default, PROC FREQ displays these two-way crosstabulation tables in table cell format. Alternatively, if you specify the CROSSLIST option, PROC FREQ displays the two-way crosstabulation tables in ODS column format. If you specify the LIST option, PROC FREQ displays multiway tables in list format, which presents the entire multiway crosstabulation in a single table.

119 Displayed Output 2325 Crosstabulation Tables By default, PROC FREQ displays two-way crosstabulation tables in table cell format. The row variable values are listed down the side of the table, the column variable values are listed across the top of the table, and each row and column variable level combination forms a table cell. Each cell of a crosstabulation table can contain the following information: Frequency, which is the number of observations in the table cell. (The NOFREQ option suppresses this information.) Expected frequency under the hypothesis of independence, if you specify the EXPECTED option Deviation of the cell frequency from the expected value, if you specify the DEVIATION option Cell Chi-Square, which is the cell s contribution to the total chi-square statistic, if you specify the CELLCHI2 option Tot Pct, which is the cell s percentage of the total multiway table frequency, for n-way tables when n > 2, if you specify the TOTPCT option Percent, which is the cell s percentage of the total (two-way table) frequency. (The NOPER- CENT option suppresses this information.) Row Pct, or the row percentage, which is the cell s percentage of the total frequency for its row. (The NOROW option suppresses this information.) Col Pct, or column percentage, which is the cell s percentage of the total frequency for its column. (The NOCOL option suppresses this information.) Cumulative Col%, or cumulative column percentage, if you specify the CUMCOL option The table also displays the Frequency Missing, which is the number of observations with missing values. CROSSLIST Tables If you specify the CROSSLIST option, PROC FREQ displays two-way crosstabulation tables in ODS column format. The CROSSLIST column format is different from the default crosstabulation table cell format, but the CROSSLIST table provides the same information (frequencies, percentages, and other statistics) as the default crosstabulation table. In the CROSSLIST table format, the rows of the display correspond to the crosstabulation table cells, and the columns of the display correspond to descriptive statistics such as frequencies and percentages. Each table cell is identified by the values of its TABLES row and column variable levels, with all column variable levels listed within each row variable level. The CROSSLIST table also provides row totals, column totals, and overall table totals.

120 2326 Chapter 35: The FREQ Procedure For a crosstabulation table in CROSSLIST format, PROC FREQ displays the following information: the row variable name and values the column variable name and values Frequency, which is the number of observations in the table cell. (The NOFREQ option suppresses this information.) Expected cell frequency under the hypothesis of independence, if you specify the EXPECTED option Deviation of the cell frequency from the expected value, if you specify the DEVIATION option Cell Chi-Square, which is the cell s contribution to the total chi-square statistic, if you specify the CELLCHI2 option Total Percent, which is the cell s percentage of the total multiway table frequency, for n-way tables when n > 2, if you specify the TOTPCT option Percent, which is the cell s percentage of the total (two-way table) frequency. (The NOPER- CENT option suppresses this information.) Row Percent, which is the cell s percentage of the total frequency for its row. (The NOROW option suppresses this information.) Column Percent, the cell s percentage of the total frequency for its column. (The NOCOL option suppresses this information.) The table also displays the Frequency Missing, which is the number of observations with missing values. LIST Tables If you specify the LIST option in the TABLES statement, PROC FREQ displays multiway tables in a list format rather than as crosstabulation tables. The LIST option displays the entire multiway table in one table, instead of displaying a separate two-way table for each stratum. The LIST option is not available when you also request statistical options. Unlike the default crosstabulation output, the LIST output does not display row percentages, column percentages, and optional information such as expected frequencies and cell chi-squares. For a multiway table in list format, PROC FREQ displays the following information: the variable names and values Frequency, which is the number of observations in the level (with the indicated variable values) Percent, which is the level s percentage of the total number of observations. (The NOPER- CENT option suppresses this information.)

121 Displayed Output 2327 Cumulative Frequency, which is the accumulated frequency of the level and all other levels listed above it in the table. The last cumulative frequency in the table is the total number of nonmissing observations. (The NOCUM option suppresses this information.) Cumulative Percent, which is the accumulated percentage of the level and all other levels listed above it in the table. (The NOCUM or the NOPERCENT option suppresses this information.) The table also displays the Frequency Missing, which is the number of observations with missing values. Statistics for Multiway Tables PROC FREQ computes statistical tests and measures for crosstabulation tables, depending on which statements and options you specify. You can suppress the display of all these results by specifying the NOPRINT option in the PROC statement. With any of the following information, PROC FREQ also displays the Sample Size and the Frequency Missing. If you specify the SCOROUT option, PROC FREQ displays the Row Scores and Column Scores that it uses for statistical computations. The Row Scores table displays the row variable values and the Score corresponding to each value. The Column Scores table displays the column variable values and the corresponding Scores. PROC FREQ also identifies the score type used to compute the row and column scores. You can specify the score type with the SCORES= option in the TABLES statement. If you specify the CHISQ option, PROC FREQ displays the following statistics for each two-way table: Pearson Chi-Square, Likelihood-Ratio Chi-Square, Continuity-Adjusted Chi- Square (for 2 2 tables), Mantel-Haenszel Chi-Square, the Phi Coefficient, the Contingency Coefficient, and Cramer s V. For each test statistic, PROC FREQ also displays the degrees of freedom (DF) and the probability value (Prob). If you specify the CHISQ option for 2 2 tables, PROC FREQ also displays Fisher s exact test. The test output includes the cell (1,1) frequency (F), the exact left-sided and right-sided probability values, the table probability (P), and the exact two-sided probability value. If you specify the FISHER option in the TABLES statement (or, equivalently, the FISHER option in the EXACT statement), PROC FREQ displays Fisher s exact test for tables larger than 2 2. The test output includes the table probability (P) and the probability value. In addition, PROC FREQ displays the CHISQ output listed earlier, even if you do not also specify the CHISQ option. If you specify the PCHI, LRCHI, or MHCHI option in the EXACT statement, PROC FREQ also displays the corresponding exact test: Pearson Chi-Square, Likelihood-Ratio Chi-Square, or Mantel-Haenszel Chi-Square, respectively. The test output includes the test statistic, the degrees of freedom (DF), and the asymptotic and exact probability values. If you also specify the POINT option in the EXACT statement, PROC FREQ displays the point probability for each exact test requested. If you specify the CHISQ option in the EXACT statement, PROC FREQ displays exact probability values for all three of these chi-square tests.

122 2328 Chapter 35: The FREQ Procedure If you specify the MEASURES option, PROC FREQ displays the following statistics and their asymptotic standard errors (ASE) for each two-way table: Gamma, Kendall s Tau-b, Stuart s Tau-c, Somers D.C jr/, Somers D.RjC /, Pearson Correlation, Spearman Correlation, Lambda Asymmetric.C jr/, Lambda Asymmetric.RjC /, Lambda Symmetric, Uncertainty Coefficient.C jr/, Uncertainty Coefficient.RjC /, and Uncertainty Coefficient Symmetric. If you specify the CL option, PROC FREQ also displays confidence limits for these measures. If you specify the PLCORR option, PROC FREQ displays the tetrachoric correlation for 2 2 tables or the polychoric correlation for larger tables. In addition, PROC FREQ displays the MEASURES output listed earlier, even if you do not also specify the MEASURES option. If you specify the GAMMA, KENTB, STUTC, SMDCR, SMDRC, PCORR, or SCORR option in the TEST statement, PROC FREQ displays asymptotic tests for Gamma, Kendall s Tau-b, Stuart s Tau-c, Somers D.C jr/, Somers D.RjC /, the Pearson Correlation, or the Spearman Correlation, respectively. If you specify the MEASURES option in the TEST statement, PROC FREQ displays all these asymptotic tests. The test output includes the statistic, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the one-sided and two-sided probability values. If you specify the KENTB, STUTC, SMDCR, SMDRC, PCORR, or SCORR option in the EXACT statement, PROC FREQ displays asymptotic and exact tests for the corresponding measure of association: Kendall s Tau-b, Stuart s Tau-c, Somers D.C jr/, Somers D.RjC /, the Pearson Correlation, or the Spearman correlation, respectively. The test output includes the correlation, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the asymptotic and exact one-sided and two-sided probability values. If you also specify the POINT option in the EXACT statement, PROC FREQ displays the point probability for each exact test requested. If you specify the RISKDIFF option for 2 2 tables, PROC FREQ displays the Column 1 and Column 2 Risk Estimates. For each column, PROC FREQ displays the Row 1 Risk, Row 2 Risk, Total Risk, and Risk Difference, together with their asymptotic standard errors (ASE) and Asymptotic Confidence Limits. PROC FREQ also displays Exact Confidence Limits for the Row 1 Risk, Row 2 Risk, and Total Risk. If you specify the RISKDIFF option in the EXACT statement, PROC FREQ provides unconditional Exact Confidence Limits for the Risk Difference. If you request a noninferiority or superiority test for the proportion difference by specifying the NONINF or SUP riskdiff-option, and if you specify METHOD=HA (Hauck-Anderson), METHOD=FM (Farrington-Manning), or METHOD=WALD (Wald), PROC FREQ displays the following information: the Proportion Difference, the test ASE (H0, Sample, Sample H-A, or FM, depending on the method you specify), the test statistic Z, the probability value, the Noninferiority or Superiority Limit, and the test-based Confidence Limits. If you specify METHOD=SCORE (Newcombe score), PROC FREQ displays the Proportion Difference, the Noninferiority or Superiority Limit, and the score Confidence Limits. If you request an equivalence test for the proportion difference by specifying the EQUIV riskdiff-option, and if you specify METHOD=HA (Hauck-Anderson), METHOD=FM (Farrington-Manning), or METHOD=WALD (Wald), PROC FREQ displays the following information: the Proportion Difference and the test ASE (H0, Sample, Sample H-A, or FM, depending on the method you specify). PROC FREQ displays a two one-sided test (TOST) for

123 Displayed Output 2329 equivalence, which includes test statistics (Z) and probability values for the Lower and Upper tests, together with the Overall probability value. PROC FREQ also displays the Equivalence Limits and the test-based Confidence Limits. If you specify METHOD=SCORE (Newcombe score), PROC FREQ displays the Proportion Difference, the Equivalence Limits, and the score Confidence Limits. If you request an equality test for the proportion difference by specifying the EQUAL riskdiffoption, PROC FREQ displays the following information: the Proportion Difference and the test ASE (H0 or Sample), the test statistic Z, the One-Sided probability value (Pr > Z or Pr < Z), and the Two-Sided probability value, Pr > Z. If you specify the MEASURES option or the RELRISK option for 2 2 tables, PROC FREQ displays Estimates of the Relative Risk for Case-Control and Cohort studies, together with their Confidence Limits. These measures are also known as the Odds Ratio and the Column 1 and 2 Relative Risks. If you specify the OR option in the EXACT statement, PROC FREQ also displays Exact Confidence Limits for the Odds Ratio. If you specify the TREND option, PROC FREQ displays the Cochran-Armitage Trend Test for tables that are 2 C or R 2. For this test, PROC FREQ gives the Statistic (Z) and the one-sided and two-sided probability values. If you specify the TREND option in the EXACT statement, PROC FREQ also displays the exact one-sided and two-sided probability values for this test. If you specify the POINT option with the TREND option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic. If you specify the JT option, PROC FREQ displays the Jonckheere-Terpstra Test, showing the Statistic (JT), the standardized test statistic (Z), and the one-sided and two-sided probability values. If you specify the JT option in the EXACT statement, PROC FREQ also displays the exact one-sided and two-sided probability values for this test. If you specify the POINT option with the JT option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic. If you specify the AGREE option and the PRINTKWT option, PROC FREQ displays the Kappa Coefficient Weights for square tables greater than 2 2. If you specify the AGREE option, for two-way tables PROC FREQ displays McNemar s Test and the Simple Kappa Coefficient for 2 2 tables. For square tables larger than 2 2, PROC FREQ displays Bowker s Test of Symmetry, the Simple Kappa Coefficient, and the Weighted Kappa Coefficient. For McNemar s Test and Bowker s Test of Symmetry, PROC FREQ displays the Statistic (S), the degrees of freedom (DF), and the probability value (Pr > S). If you specify the MCNEM option in the EXACT statement, PROC FREQ also displays the exact probability value for McNemar s test. If you specify the POINT option with the MCNEM option in the EXACT statement, PROC FREQ displays the exact point probability for the test statistic. For the simple and weighted kappa coefficients, PROC FREQ displays the kappa values, asymptotic standard errors (ASE), and Confidence Limits. If you specify the KAPPA or WTKAP option in the TEST statement, PROC FREQ displays asymptotic tests for the simple kappa coefficient or the weighted kappa coefficient, respectively. If you specify the AGREE option in the TEST statement, PROC FREQ displays both these asymptotic tests. The test output includes the kappa coefficient, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the one-sided and two-sided probability values.

124 2330 Chapter 35: The FREQ Procedure If you specify the KAPPA or WTKAP option in the EXACT statement, PROC FREQ displays asymptotic and exact tests for the simple kappa coefficient or the weighted kappa coefficient, respectively. The test output includes the kappa coefficient, its asymptotic standard error (ASE), Confidence Limits, the ASE under the null hypothesis H0, the standardized test statistic (Z), and the asymptotic and exact one-sided and two-sided probability values. If you specify the POINT option in the EXACT statement, PROC FREQ displays the point probability for each exact test requested. If you specify the MC option in the EXACT statement, PROC FREQ displays Monte Carlo estimates for all exact p-values requested by statistic-options in the EXACT statement. The Monte Carlo output includes the p-value Estimate, its Confidence Limits, the Number of Samples used to compute the Monte Carlo estimate, and the Initial Seed for random number generation. If you specify the AGREE option, for multiple strata PROC FREQ displays Overall Simple and Weighted Kappa Coefficients, with their asymptotic standard errors (ASE) and Confidence Limits. PROC FREQ also displays Tests for Equal Kappa Coefficients, giving the Chi-Squares, degrees of freedom (DF), and probability values (Pr > ChiSq) for the Simple Kappa and Weighted Kappa. For multiple strata of 2 2 tables, PROC FREQ displays Cochran s Q, giving the Statistic (Q), the degrees of freedom (DF), and the probability value (Pr > Q). If you specify the CMH option, PROC FREQ displays Cochran-Mantel-Haenszel Statistics for the following three alternative hypotheses: Nonzero Correlation, Row Mean Scores Differ (ANOVA Statistic), and General Association. For each of these statistics, PROC FREQ gives the degrees of freedom (DF) and the probability value (Prob). If you specify the MANTELFLEISS option, PROC FREQ displays the Mantel-Fleiss Criterion for 2 2 tables. For 2 2 tables, PROC FREQ also displays Estimates of the Common Relative Risk for Case-Control and Cohort studies, together with their confidence limits. These include both Mantel-Haenszel and Logit stratum-adjusted estimates of the common Odds Ratio, Column 1 Relative Risk, and Column 2 Relative Risk. Also for 2 2 tables, PROC FREQ displays the Breslow-Day Test for Homogeneity of the Odds Ratios. For this test, PROC FREQ gives the Chi-Square, the degrees of freedom (DF), and the probability value (Pr > ChiSq). If you specify the CMH option in the TABLES statement and also specify the COMOR option in the EXACT statement, PROC FREQ displays exact confidence limits for the Common Odds Ratio for multiple strata of 2 2 tables. PROC FREQ also displays the Exact Test of H0: Common Odds Ratio = 1. The test output includes the Cell (1,1) Sum (S), Mean of S Under H0, One-sided Pr <= S, and Point Pr = S. PROC FREQ also provides exact two-sided probability values for the test, computed according to the following three methods: 2 * One-sided, Sum of probabilities <= Point probability, and Pr >= S - Mean. If you specify the CMH option in the TABLES statement and also specify the EQOR option in the EXACT statement, PROC FREQ computes Zelen s exact test for equal odds ratios for h 2 2 tables. PROC FREQ displays Zelen s test along with the asymptotic Breslow-Day test produced by the CMH option. PROC FREQ displays the test statistic, Zelen s Exact Test (P), and the probability value, Exact Pr <= P. If you specify the GAILSIMON option in the TABLES statement for a multiway 2 2 tables, PROC FREQ displays the Gail-Simon test for qualitative interactions. The display include the following statistics and their p-values: Q+ (Positive Risk Differences), Q- (Negative Risk Differences), and Q (Two-Sided).

125 ODS Table Names 2331 ODS Table Names PROC FREQ assigns a name to each table that it creates. You can use these names to refer to tables when you use the Output Delivery System (ODS) to select tables and create output data sets. For more information about ODS, see Chapter 20, Using the Output Delivery System. Table lists the ODS table names together with their descriptions and the options required to produce the tables. Note that the ALL option in the TABLES statement invokes the CHISQ, MEASURES, and CMH options. Table ODS Tables Produced by PROC FREQ ODS Table Name Description Statement Option BinomialCLs Binomial confidence limits TABLES BINOMIAL(AC J W) BinomialEquiv Binomial equivalence analysis TABLES BINOMIAL(EQUIV) BinomialEquivLimits Binomial equivalence limits TABLES BINOMIAL(EQUIV) BinomialEquivTest Binomial equivalence test TABLES BINOMIAL(EQUIV) BinomialNoninf Binomial noninferiority test TABLES BINOMIAL(NONINF) BinomialProp Binomial proportion TABLES BINOMIAL BinomialPropTest Binomial proportion test TABLES BINOMIAL BinomialSup Binomial superiority test TABLES BINOMIAL(SUP) BreslowDayTest Breslow-Day test TABLES CMH (h 2 2 tables) CMH Cochran-Mantel-Haenszel TABLES CMH statistics ChiSq Chi-square tests TABLES CHISQ CochransQ Cochran s Q TABLES AGREE (h 2 2 tables) ColScores Column scores TABLES SCOROUT CommonOdds- Exact confidence limits EXACT COMOR RatioCl for the common odds ratio (h 2 2 tables) CommonOdds- Common odds ratio exact test EXACT COMOR RatioTest (h 2 2 tables) CommonRelRisks Common relative risks TABLES CMH (h 2 2 tables) CrossList Crosstabulation table TABLES CROSSLIST in column format (n-way table, n > 1) CrossTabFreqs Crosstabulation table TABLES (n-way table, n > 1) EqualKappaTest Test for equal simple kappas TABLES AGREE (h 2 2 tables) EqualKappaTests Tests for equal kappas TABLES AGREE (h r r, r > 2) EqualOddsRatios Tests for equal odds ratios EXACT EQOR (h 2 2 tables) GailSimon Gail-Simon test TABLES GAILSIMON (h 2 2 tables) FishersExact Fisher s exact test EXACT FISHER or TABLES FISHER or EXACT or TABLES CHISQ (2 2 tables)

126 2332 Chapter 35: The FREQ Procedure Table continued ODS Table Name Description Statement Option FishersExactMC Monte Carlo estimates EXACT FISHER / MC for Fisher s exact test Gamma Gamma TEST GAMMA GammaTest Gamma test TEST GAMMA JTTest Jonckheere-Terpstra test TABLES JT JTTestMC Monte Carlo estimates for EXACT JT / MC Jonckheere-Terpstra exact test KappaStatistics Kappa statistics TABLES AGREE, no TEST or EXACT (r r tables, r > 2) KappaWeights Kappa weights TABLES AGREE and PRINTKWT List List format multiway table TABLES LIST LRChiSq Likelihood-ratio chi-square EXACT LRCHI exact test LRChiSqMC Monte Carlo exact test for EXACT LRCHI / MC likelihood-ratio chi-square MantelFleiss Mantel-Fleiss criterion TABLES CMH(MF) (h 2 2 tables) McNemarsTest McNemar s test TABLES AGREE (2 2 tables) Measures Measures of association TABLES MEASURES MHChiSq Mantel-Haenszel chi-square EXACT MHCHI exact test MHChiSqMC Monte Carlo exact test for EXACT MHCHI / MC Mantel-Haenszel chi-square NLevels Number of variable levels PROC NLEVELS OddsRatioCL Exact confidence limits EXACT OR (2 2 tables) for the odds ratio OneWayChiSq One-way chi-square test TABLES CHISQ (one-way tables) OneWayChiSqMC Monte Carlo exact test for EXACT CHISQ / MC one-way chi-square (one-way tables) OneWayFreqs One-way frequencies PROC (no TABLES stmt) or TABLES (one-way table) OverallKappa Overall simple kappa TABLES AGREE (h 2 2 tables) OverallKappas Overall kappa coefficients TABLES AGREE (h r r, r > 2) PdiffEquiv Equivalence analysis TABLES RISKDIFF(EQUIV) for the proportion difference (2 2 tables) PdiffEquivLimits Equivalence limits TABLES RISKDIFF(EQUIV) for the proportion difference (2 2 tables) PdiffEquivTest Equivalence test TABLES RISKDIFF(EQUIV) for the proportion difference (2 2 tables)

127 ODS Table Names 2333 Table continued ODS Table Name Description Statement Option PdiffNoninf Noninferiority test TABLES RISKDIFF(NONINF) for the proportion difference (2 2 tables) PdiffSup Superiority test TABLES RISKDIFF(SUP) for the proportion difference (2 2 tables) PdiffTest Proportion difference test TABLES RISKDIFF(EQUAL) (2 2 tables) PearsonChiSq Pearson chi-square exact test EXACT PCHI PearsonChiSqMC Monte Carlo exact test for EXACT PCHI / MC Pearson chi-square PearsonCorr Pearson correlation TEST PCORR or EXACT PCORR PearsonCorrMC Monte Carlo exact test for EXACT PCORR / MC Pearson correlation PearsonCorrTest Pearson correlation test TEST PCORR or EXACT PCORR RelativeRisks Relative risk estimates TABLES RELRISK or MEASURES (2 2 tables) RiskDiffCol1 Column 1 risk estimates TABLES RISKDIFF (2 2 tables) RiskDiffCol2 Column 2 risk estimates TABLES RISKDIFF (2 2 tables) RowScores Row scores TABLES SCOROUT SimpleKappa Simple kappa coefficient TEST KAPPA or EXACT KAPPA SimpleKappaMC Monte Carlo exact test for EXACT KAPPA / MC simple kappa SimpleKappaTest Simple kappa test TEST KAPPA or EXACT KAPPA SomersDCR Somers D.C jr/ TEST SMDCR SomersDCRTest Somers D.C jr/ test TEST SMDCR SomersDRC Somers D.RjC / TEST SMDRC SomersDRCTest Somers D.RjC / test TEST SMDRC SpearmanCorr Spearman correlation TEST SCORR or EXACT SCORR SpearmanCorrMC Monte Carlo exact test for EXACT SCORR / MC Spearman correlation SpearmanCorrTest Spearman correlation test TEST SCORR or EXACT SCORR SymmetryTest Test of symmetry TABLES AGREE TauB Kendall s tau-b TEST KENTB TauBTest Kendall s tau-b test TEST KENTB TauC Stuart s tau-c TEST STUTC TauCTest Stuart s tau-c test TEST STUTC TrendTest Cochran-Armitage trend test TABLES TREND

128 2334 Chapter 35: The FREQ Procedure Table continued ODS Table Name Description Statement Option TrendTestMC Monte Carlo exact test EXACT TREND / MC for trend WeightedKappa Weighted kappa TEST WTKAP or EXACT WTKAP WeightedKappaMC Monte Carlo exact test for EXACT WTKAP / MC weighted kappa WeightedKappaTest Weighted kappa test TEST WTKAP or EXACT WTKAP ODS Graphics PROC FREQ assigns a name to each graph that it creates with ODS Graphics. You can use these names to refer to the graphs. Table lists the names of the graphs that PROC FREQ generates together with their descriptions and the options that are required to produce the graphs. To request graphics with PROC FREQ, you must first enable ODS Graphics by specifying the ODS GRAPHICS ON statement. See Chapter 21, Statistical Graphics Using ODS, for more information. When you have enabled ODS Graphics, you can request specific plots with the PLOTS= option in the TABLES statement. If you do not specify the PLOTS= option but have enabled ODS Graphics, then PROC FREQ produces all plots that are associated with the analyses that you request. Table ODS Graphics Produced by PROC FREQ ODS Graph Name Plot Description TABLES Statement Option CumFreqPlot Cumulative frequency plot One-way table request DeviationPlot Deviation plot CHISQ and a one-way table request FreqPlot Frequency plot Any table request KappaPlot Kappa plot AGREE (h r r table) ORPlot Odds ratio plot MEASURES or RELRISK (h 2 2 table) RelRiskPlot Relative risk plot MEASURES or RELRISK (h 2 2 table) RiskDiffPlot Risk difference plot RISKDIFF (h 2 2 table) WtKappaPlot Weighted kappa plot AGREE (h r r table, r > 2)

129 Examples: FREQ Procedure 2335 Examples: FREQ Procedure Example 35.1: Output Data Set of Frequencies The eye and hair color of children from two different regions of Europe are recorded in the data set Color. Instead of recording one observation per child, the data are recorded as cell counts, where the variable Count contains the number of children exhibiting each of the 15 eye and hair color combinations. The data set does not include missing combinations. The following DATA step statements create the SAS data set Color: data Color; input Region Eyes $ Hair $ label Eyes ='Eye Color' Hair ='Hair Color' Region='Geographic Region'; datalines; 1 blue fair 23 1 blue red 7 1 blue medium 24 1 blue dark 11 1 green fair 19 1 green red 7 1 green medium 18 1 green dark 14 1 brown fair 34 1 brown red 5 1 brown medium 41 1 brown dark 40 1 brown black 3 2 blue fair 46 2 blue red 21 2 blue medium 44 2 blue dark 40 2 blue black 6 2 green fair 50 2 green red 31 2 green medium 37 2 green dark 23 2 brown fair 56 2 brown red 42 2 brown medium 53 2 brown dark 54 2 brown black 13 ; The following PROC FREQ statements read the Color data set and create an output data set that contains the frequencies, percentages, and expected cell frequencies of the two-way table of Eyes by Hair. The TABLES statement requests three tables: a frequency table for Eyes, a frequency table for Hair, and a crosstabulation table for Eyes by Hair. The OUT= option creates the FreqCount data set, which contains the crosstabulation table frequencies. The OUTEXPECT option outputs the expected table cell frequencies to FreqCount, and the SPARSE option includes zero cell frequencies in the output data set. The WEIGHT statement specifies that the variable Count contains the observation weights. These statements create Output through Output proc freq data=color; tables Eyes Hair Eyes*Hair / out=freqcount outexpect sparse; weight Count; title 'Eye and Hair Color of European Children'; run; proc print data=freqcount noobs; title2 'Output Data Set from PROC FREQ'; run;

130 2336 Chapter 35: The FREQ Procedure Output displays the two frequency tables produced by PROC FREQ: one showing the distribution of eye color, and one showing the distribution of hair color. By default, PROC FREQ lists the variables values in alphabetical order. The Eyes*Hair specification produces a crosstabulation table, shown in Output , with eye color defining the table rows and hair color defining the table columns. A zero cell frequency for green eyes and black hair indicates that this eye and hair color combination does not occur in the data. The output data set FreqCount (Output ) contains frequency counts and percentages for the last table requested in the TABLES statement, Eyes by Hair. Because the SPARSE option is specified, the data set includes the observation with a zero frequency. The variable Expected contains the expected frequencies, as requested by the OUTEXPECT option. Output Frequency Tables Eye and Hair Color of European Children The FREQ Procedure Eye Color Cumulative Cumulative Eyes Frequency Percent Frequency Percent blue brown green Hair Color Cumulative Cumulative Hair Frequency Percent Frequency Percent black dark fair medium red

131 Example 35.1: Output Data Set of Frequencies 2337 Output Crosstabulation Table Table of Eyes by Hair Eyes(Eye Color) Hair(Hair Color) Frequency Percent Row Pct Col Pct black dark fair medium red Total blue brown green Total Output Output Data Set of Frequencies Eye and Hair Color of European Children Output Data Set from PROC FREQ Eyes Hair COUNT EXPECTED PERCENT blue black blue dark blue fair blue medium blue red brown black brown dark brown fair brown medium brown red green black green dark green fair green medium green red

132 2338 Chapter 35: The FREQ Procedure Example 35.2: Frequency Dot Plots This example produces frequency dot plots for the children s eye and hair color data from Example PROC FREQ produces plots by using ODS Graphics to create graphs as part of the procedure output. Frequency plots are available for any frequency or crosstabulation table request. You can display frequency plots as bar charts or dot plots. You can use plot-options to specify the orientation (vertical or horizontal), scale, and layout of the plots. The following PROC FREQ statements request frequency tables and dot plots. The first TABLES statement requests a one-way frequency table of Hair and a crosstabulation table of Eyes by Hair. The PLOTS= option requests frequency plots for the tables, and the TYPE=DOT plot-option specifies dot plots. By default, frequency plots are produced as bar charts. The second TABLES statement requests a crosstabulation table of Region by Hair and a frequency dot plot for this table. The SCALE=PERCENT plot-option plots percentages instead of frequency counts. SCALE=LOG and SCALE=SQRT plot-options are also available to plot log frequencies and square roots of frequencies, respectively. The ORDER=FREQ option in the PROC FREQ statement orders the variable levels by frequency. This order applies to the frequency and crosstabulation table displays and also to the corresponding frequency plots. Before requesting plots, you must enable ODS Graphics with the ODS GRAPHICS ON statement. ods graphics on; proc freq data=color order=freq; tables Hair Eyes*Hair / plots=freqplot(type=dot); tables Region*Hair / plots=freqplot(type=dot scale=percent); weight Count; title 'Eye and Hair Color of European Children'; run; ods graphics off; Output , Output , and Output display the dot plots produced by PROC FREQ. By default, the orientation of dot plots is horizontal, which places the variable levels on the Y axis. You can specify the ORIENT=VERTICAL plot-option to request a vertical orientation. For two-way plots, you can use the TWOWAY= plot-option to specify the plot layout. The default layout (shown in Output and Output ) is GROUPVERTICAL. Two-way layouts STACKED and GROUPHORIZONTAL are also available.

133 Output One-Way Frequency Dot Plot Example 35.2: Frequency Dot Plots 2339

134 2340 Chapter 35: The FREQ Procedure Output Two-Way Frequency Dot Plot

135 Example 35.3: Chi-Square Goodness-of-Fit Tests 2341 Output Two-Way Percent Dot Plot Example 35.3: Chi-Square Goodness-of-Fit Tests This example examines whether the children s hair color (from Example 35.1) has a specified multinomial distribution for the two geographical regions. The hypothesized distribution of hair color is 30% fair, 12% red, 30% medium, 25% dark, and 3% black. In order to test the hypothesis for each region, the data are first sorted by Region. Then the FREQ procedure uses a BY statement to produce a separate table for each BY group (Region). The option ORDER=DATA orders the variable values (hair color) in the frequency table by their order in the input data set. The TABLES statement requests a frequency table for hair color, and the option NOCUM suppresses the display of the cumulative frequencies and percentages. The CHISQ option requests a chi-square goodness-of-fit test for the frequency table of Hair. The TESTP= option specifies the hypothesized (or test) percentages for the chi-square test; the number of percentages listed equals the number of table levels, and the percentages sum to 100%. The TESTP= percentages are listed in the same order as the corresponding variable levels appear in frequency table.

136 2342 Chapter 35: The FREQ Procedure The PLOTS= option requests a deviation plot, which is associated with the CHISQ option and displays the relative deviations from the test frequencies. The TYPE=DOT plot-option requests a dot plot instead of the default type, which is a bar chart. The ONLY global-plot-option requests that PROC FREQ produce only the deviation plot. By default, PROC FREQ produces all plots associated with the requested analyses. A frequency plot is associated with a one-way table request but is not produced in this example because ONLY is specified with the DEVIATIONPLOT request. Note that ODS Graphics must be enabled before requesting plots. These statements produce Output through Output proc sort data=color; by Region; run; ods graphics on; proc freq data=color order=data; tables Hair / nocum chisq testp=( ) plots(only)=deviationplot(type=dot); weight Count; by Region; title 'Hair Color of European Children'; run; ods graphics off; Output Frequency Table and Chi-Square Test for Region 1 Hair Color of European Children Geographic Region= The FREQ Procedure Hair Color Test Hair Frequency Percent Percent fair red medium dark black Geographic Region= Chi-Square Test for Specified Proportions Chi-Square DF 4 Pr > ChiSq

137 Example 35.3: Chi-Square Goodness-of-Fit Tests 2343 Output shows the frequency table and chi-square test for Region 1. The frequency table lists the variable values (hair color) in the order in which they appear in the data set. The Test Percent column lists the hypothesized percentages for the chi-square test. Always check that you have ordered the TESTP= percentages to correctly match the order of the variable levels. Output shows the deviation plot for Region 1, which displays the relative deviations from the hypothesized values. The relative deviation for a level is the difference between the observed and hypothesized (test) percentage divided by the test percentage. You can suppress the chi-square p-value that is displayed by default in the deviation plot by specifying the NOSTATS plot-option. Output Deviation Plot for Region 1

138 2344 Chapter 35: The FREQ Procedure Output and Output show the results for Region 2. PROC FREQ computes a chisquare statistic for each region. The chi-square statistic is significant at the 0.05 level for Region 2 (p=0.0003) but not for Region 1. This indicates a significant departure from the hypothesized percentages in Region 2. Output Frequency Table and Chi-Square Test for Region 2 Hair Color of European Children Geographic Region= The FREQ Procedure Hair Color Test Hair Frequency Percent Percent fair red medium dark black Geographic Region= Chi-Square Test for Specified Proportions Chi-Square DF 4 Pr > ChiSq

139 Example 35.4: Binomial Proportions 2345 Output Deviation Plot for Region 2 Example 35.4: Binomial Proportions In this example, PROC FREQ computes binomial proportions, confidence limits, and tests. The example uses the eye and hair color data from Example By default, PROC FREQ computes the binomial proportion as the proportion of observations in the first level of the one-way table. You can designate a different level by using the LEVEL= binomial-option. The following PROC FREQ statements compute the proportion of children with brown eyes (from the data set in Example 35.1) and test the null hypothesis that the population proportion equals 50%. These statements also compute an equivalence for the proportion of children with fair hair. The first TABLES statement requests a one-way frequency table for the variable Eyes. The BINO- MIAL option requests the binomial proportion, confidence limits, and test. PROC FREQ computes the proportion with Eyes = brown, which is the first level displayed in the table. The AC, WILSON, and EXACT binomial-options request the following confidence limits types: Agresti-Coull, Wilson (score), and exact (Clopper-Pearson). By default, PROC FREQ provides Wald and exact (Clopper-

140 2346 Chapter 35: The FREQ Procedure Pearson) confidence limits for the binomial proportion. The BINOMIAL option also produces an asymptotic Wald test that the proportion equals 0.5. You can specify a different test proportion with the P= binomial-option. The ALPHA=0.1 option specifies that D 10%, which produces 90% confidence limits. The second TABLES statement requests a one-way frequency table for the variable Hair. The BINOMIAL option requests the proportion for the first level, Hair = fair. The EQUIV binomialoption requests an equivalence test for the binomial proportion. The P=.28 option specifies 0.28 as the null hypothesis proportion, and the MARGIN=.1 option specifies 0.1 as the equivalence test margin. proc freq data=color order=freq; tables Eyes / binomial(ac wilson exact) alpha=.1; tables Hair / binomial(equiv p=.28 margin=.1); weight Count; title 'Hair and Eye Color of European Children'; run; Output displays the results for eye color, and Output displays the results for hair color. Output Binomial Proportion for Eye Color Hair and Eye Color of European Children The FREQ Procedure Eye Color Cumulative Cumulative Eyes Frequency Percent Frequency Percent brown blue green Binomial Proportion for Eyes = brown Proportion ASE Type 90% Confidence Limits Wilson Agresti-Coull Clopper-Pearson (Exact) Test of H0: Proportion = 0.5 ASE under H Z One-sided Pr < Z Two-sided Pr > Z

141 Example 35.4: Binomial Proportions 2347 The frequency table in Output displays the values of Eyes in order of descending frequency count. PROC FREQ computes the proportion of children in the first level displayed in the frequency table, Eyes = brown. Output displays the binomial proportion confidence limits and test. The confidence limits are 90% confidence limits. If you do not specify the ALPHA= option, PROC FREQ computes 95% confidence limits by default. Because the value of Z is less than zero, PROC FREQ displays the a left-sided p-value (0.0019). This small p-value supports the alternative hypothesis that the true value of the proportion of children with brown eyes is less than 50%. Output displays the equivalence test results produced by the second TABLES statement. The null hypothesis proportion is 0.28 and the equivalence margins are 0.1 and 0.1, which yield equivalence limits of 0.18 and PROC FREQ provides two one-sided tests (TOST) for equivalence. The small p-value indicates rejection of the null hypothesis in favor of the alternative that the proportion is equivalent to the null value. Output Binomial Proportion for Hair Color Hair Color Cumulative Cumulative Hair Frequency Percent Frequency Percent fair medium dark red black Equivalence Analysis H0: P - p0 <= Lower Margin or >= Upper Margin Ha: Lower Margin < P - p0 < Upper Margin p0 = 0.28 Lower Margin = -0.1 Upper Margin = 0.1 Proportion ASE (Sample) Two One-Sided Tests (TOST) Test Z P-Value Lower Margin Pr > Z <.0001 Upper Margin Pr < Z <.0001 Overall <.0001 Equivalence Limits 90% Confidence Limits

142 2348 Chapter 35: The FREQ Procedure Example 35.5: Analysis of a 2x2 Contingency Table This example computes chi-square tests and Fisher s exact test to compare the probability of coronary heart disease for two types of diet. It also estimates the relative risks and computes exact confidence limits for the odds ratio. The data set FatComp contains hypothetical data for a case-control study of high fat diet and the risk of coronary heart disease. The data are recorded as cell counts, where the variable Count contains the frequencies for each exposure and response combination. The data set is sorted in descending order by the variables Exposure and Response, so that the first cell of the 2 2 table contains the frequency of positive exposure and positive response. The FORMAT procedure creates formats to identify the type of exposure and response with character values. proc format; value ExpFmt 1='High Cholesterol Diet' 0='Low Cholesterol Diet'; value RspFmt 1='Yes' 0='No'; run; data FatComp; input Exposure Response Count; label Response='Heart Disease'; datalines; ; proc sort data=fatcomp; by descending Exposure descending Response; run; In the following PROC FREQ statements, ORDER=DATA option orders the contingency table values by their order in the input data set. The TABLES statement requests a two-way table of Exposure by Response. The CHISQ option produces several chi-square tests, while the RELRISK option produces relative risk measures. The EXACT statement requests the exact Pearson chi-square test and exact confidence limits for the odds ratio. proc freq data=fatcomp order=data; format Exposure ExpFmt. Response RspFmt.; tables Exposure*Response / chisq relrisk; exact pchi or; weight Count; title 'Case-Control Study of High Fat/Cholesterol Diet'; run;

143 Example 35.5: Analysis of a 2x2 Contingency Table 2349 The contingency table in Output displays the variable values so that the first table cell contains the frequency for the first cell in the data set (the frequency of positive exposure and positive response). Output Contingency Table Case-Control Study of High Fat/Cholesterol Diet The FREQ Procedure Table of Exposure by Response Exposure Response(Heart Disease) Frequency Percent Row Pct Col Pct Yes No Total High Cholesterol Diet Low Cholesterol Diet Total Output displays the chi-square statistics. Because the expected counts in some of the table cells are small, PROC FREQ gives a warning that the asymptotic chi-square tests might not be appropriate. In this case, the exact tests are appropriate. The alternative hypothesis for this analysis states that coronary heart disease is more likely to be associated with a high fat diet, so a one-sided test is desired. Fisher s exact right-sided test analyzes whether the probability of heart disease in the high fat group exceeds the probability of heart disease in the low fat group; because this p-value is small, the alternative hypothesis is supported. The odds ratio, displayed in Output , provides an estimate of the relative risk when an event is rare. This estimate indicates that the odds of heart disease is 8.25 times higher in the high fat diet group; however, the wide confidence limits indicate that this estimate has low precision.

144 2350 Chapter 35: The FREQ Procedure Output Chi-Square Statistics Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Continuity Adj. Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V WARNING: 50% of the cells have expected counts less than 5. (Asymptotic) Chi-Square may not be a valid test. Pearson Chi-Square Test Chi-Square DF 1 Asymptotic Pr > ChiSq Exact Pr >= ChiSq Fisher's Exact Test Cell (1,1) Frequency (F) 11 Left-sided Pr <= F Right-sided Pr >= F Table Probability (P) Two-sided Pr <= P Output Relative Risk Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) Cohort (Col1 Risk) Cohort (Col2 Risk) Odds Ratio (Case-Control Study) Odds Ratio Asymptotic Conf Limits 95% Lower Conf Limit % Upper Conf Limit Exact Conf Limits 95% Lower Conf Limit % Upper Conf Limit

145 Example 35.6: Output Data Set of Chi-Square Statistics 2351 Example 35.6: Output Data Set of Chi-Square Statistics This example uses the Color data from Example 35.1 to output the Pearson chi-square and the likelihood-ratio chi-square statistics to a SAS data set. The following PROC FREQ statements create a two-way table of eye color versus hair color. proc freq data=color order=data; tables Eyes*Hair / expected cellchi2 norow nocol chisq; output out=chisqdata n nmiss pchi lrchi; weight Count; title 'Chi-Square Tests for 3 by 5 Table of Eye and Hair Color'; run; proc print data=chisqdata noobs; title1 'Chi-Square Statistics for Eye and Hair Color'; title2 'Output Data Set from the FREQ Procedure'; run; The EXPECTED option displays expected cell frequencies in the crosstabulation table, and the CELLCHI2 option displays the cell contribution to the overall chi-square. The NOROW and NOCOL options suppress the display of row and column percents in the crosstabulation table. The CHISQ option produces chi-square tests. The OUTPUT statement creates the ChiSqData output data set and specifies the statistics to include. The N option requests the number of nonmissing observations, the NMISS option stores the number of missing observations, and the PCHI and LRCHI options request Pearson and likelihood-ratio chi-square statistics, respectively, together with their degrees of freedom and p-values. The preceding statements produce Output and Output The contingency table in Output displays eye and hair color in the order in which they appear in the Color data set. The Pearson chi-square statistic in Output provides evidence of an association between eye and hair color (p=0.0073). The cell chi-square values show that most of the association is due to more green-eyed children with fair or red hair and fewer with dark or black hair. The opposite occurs with the brown-eyed children. Output displays the output data set created by the OUTPUT statement. It includes one observation that contains the sample size, the number of missing values, and the chi-square statistics and corresponding degrees of freedom and p-values as in Output

146 2352 Chapter 35: The FREQ Procedure Output Contingency Table Chi-Square Tests for 3 by 5 Table of Eye and Hair Color The FREQ Procedure Table of Eyes by Hair Eyes(Eye Color) Hair(Hair Color) Frequency Expected Cell Chi-Square Percent fair red medium dark black Total blue green brown Total Output Chi-Square Statistics Statistic DF Value Prob Chi-Square Likelihood Ratio Chi-Square Mantel-Haenszel Chi-Square Phi Coefficient Contingency Coefficient Cramer's V Output Output Data Set Chi-Square Statistics for Eye and Hair Color Output Data Set from the FREQ Procedure N NMISS _PCHI_ DF_PCHI P_PCHI _LRCHI_ DF_LRCHI P_LRCHI

147 Example 35.7: Cochran-Mantel-Haenszel Statistics 2353 Example 35.7: Cochran-Mantel-Haenszel Statistics The data set Migraine contains hypothetical data for a clinical trial of migraine treatment. Subjects of both genders receive either a new drug therapy or a placebo. Their response to treatment is coded as Better or Same. The data are recorded as cell counts, and the number of subjects for each treatment and response combination is recorded in the variable Count. data Migraine; input Gender $ Treatment $ Response $ datalines; female Active Better 16 female Active Same 11 female Placebo Better 5 female Placebo Same 20 male Active Better 12 male Active Same 16 male Placebo Better 7 male Placebo Same 19 ; The following PROC FREQ statements create a multiway table stratified by Gender, where Treatment forms the rows and Response forms the columns. The CMH option produces the Cochran-Mantel- Haenszel statistics. For this stratified 2 2 table, estimates of the common relative risk and the Breslow-Day test for homogeneity of the odds ratios are also displayed. The NOPRINT option suppresses the display of the contingency tables. These statements produce Output through Output proc freq data=migraine; tables Gender*Treatment*Response / cmh; weight Count; title 'Clinical Trial for Treatment of Migraine Headaches'; run; Output Cochran-Mantel-Haenszel Statistics Clinical Trial for Treatment of Migraine Headaches The FREQ Procedure Summary Statistics for Treatment by Response Controlling for Gender Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Prob Nonzero Correlation Row Mean Scores Differ General Association For a stratified 2 2 table, the three CMH statistics displayed in Output test the same hypothesis. The significant p-value (0.004) indicates that the association between treatment and response remains strong after adjusting for gender.

148 2354 Chapter 35: The FREQ Procedure The CMH option also produces a table of relative risks, as shown in Output Because this is a prospective study, the relative risk estimate assesses the effectiveness of the new drug; the Cohort (Col1 Risk) values are the appropriate estimates for the first column (the risk of improvement). The probability of migraine improvement with the new drug is just over two times the probability of improvement with the placebo. The large p-value for the Breslow-Day test (0.2218) in Output indicates no significant gender difference in the odds ratios. Output CMH Option: Relative Risks Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits Case-Control Mantel-Haenszel (Odds Ratio) Logit Cohort Mantel-Haenszel (Col1 Risk) Logit Cohort Mantel-Haenszel (Col2 Risk) Logit Output CMH Option: Breslow-Day Test Breslow-Day Test for Homogeneity of the Odds Ratios Chi-Square DF 1 Pr > ChiSq Example 35.8: Cochran-Armitage Trend Test The data set Pain contains hypothetical data for a clinical trial of a drug therapy to control pain. The clinical trial investigates whether adverse responses increase with larger drug doses. Subjects receive either a placebo or one of four drug doses. An adverse response is recorded as Adverse= Yes ; otherwise, it is recorded as Adverse= No. The number of subjects for each drug dose and response combination is contained in the variable Count. data pain; input Dose Adverse $ datalines; 0 No 26 0 Yes 6 1 No 26 1 Yes 7 2 No 23 2 Yes 9 3 No 18 3 Yes 14 4 No 9 4 Yes 23 ;

149 Example 35.8: Cochran-Armitage Trend Test 2355 The following PROC FREQ statements provide a trend analysis. The TABLES statement requests a table of Adverse by Dose. The MEASURES option produces measures of association, and the CL option produces confidence limits for these measures. The TREND option tests for a trend across the ordinal values of the variable Dose with the Cochran-Armitage test. The EXACT statement produces exact p-values for this test, and the MAXTIME= option terminates the exact computations if they do not complete within 60 seconds. The TEST statement computes an asymptotic test for Somers D.RjC /. The PLOTS= option requests a frequency plot for the table of Adverse by Dose. By default, PROC FREQ provides a bar chart for the frequency plot. The TWOWAY=STACKED option requests a stacked layout, where the bars correspond to the column variable (Dose) values, and the row variable (Adverse) frequencies are stacked within each bar. ods graphics on; proc freq data=pain; tables Adverse*Dose / trend measures cl plots=freqplot(twoway=stacked); test smdrc; exact trend / maxtime=60; weight Count; title 'Clinical Trial for Treatment of Pain'; run; ods graphics off; Output through Output display the results of the analysis. The Col Pct values in Output show the expected increasing trend in the proportion of adverse effects with the increasing dosage (from 18.75% to 71.88%). The corresponding frequency bar chart (Output ) also shows this increasing trend. Output Contingency Table Clinical Trial for Treatment of Pain The FREQ Procedure Table of Adverse by Dose Adverse Dose Frequency Percent Row Pct Col Pct Total No Yes Total

150 2356 Chapter 35: The FREQ Procedure Output Stacked Bar Chart of Frequencies Output displays the measures of association produced by the MEASURES option. Somers D.RjC / measures the association treating the row variable (Adverse) as the response and the column variable (Dose) as a predictor. Because the asymptotic 95% confidence limits do not contain zero, this indicates a strong positive association. Similarly, the Pearson and Spearman correlation coefficients show evidence of a strong positive association, as hypothesized. The Cochran-Armitage test (Output ) supports the trend hypothesis. The small left-sided p- values for the Cochran-Armitage test indicate that the probability of the Row 1 level (Adverse= No ) decreases as Dose increases or, equivalently, that the probability of the Row 2 level (Adverse= Yes ) increases as Dose increases. The two-sided p-value tests against either an increasing or decreasing alternative. This is an appropriate hypothesis when you want to determine whether the drug has progressive effects on the probability of adverse effects but the direction is unknown.

151 Example 35.8: Cochran-Armitage Trend Test 2357 Output Measures of Association 95% Statistic Value ASE Confidence Limits Gamma Kendall's Tau-b Stuart's Tau-c Somers' D C R Somers' D R C Pearson Correlation Spearman Correlation Lambda Asymmetric C R Lambda Asymmetric R C Lambda Symmetric Uncertainty Coefficient C R Uncertainty Coefficient R C Uncertainty Coefficient Symmetric Somers' D R C Somers' D R C ASE % Lower Conf Limit % Upper Conf Limit Test of H0: Somers' D R C = 0 ASE under H Z One-sided Pr > Z <.0001 Two-sided Pr > Z <.0001 Output Trend Test Cochran-Armitage Trend Test Statistic (Z) Asymptotic Test One-sided Pr < Z <.0001 Two-sided Pr > Z <.0001 Exact Test One-sided Pr <= Z 7.237E-07 Two-sided Pr >= Z 1.324E-06

152 2358 Chapter 35: The FREQ Procedure Example 35.9: Friedman s Chi-Square Test Friedman s test is a nonparametric test for treatment differences in a randomized complete block design. Each block of the design might be a subject or a homogeneous group of subjects. If blocks are groups of subjects, the number of subjects in each block must equal the number of treatments. Treatments are randomly assigned to subjects within each block. If there is one subject per block, then the subjects are repeatedly measured once under each treatment. The order of treatments is randomized for each subject. In this setting, Friedman s test is identical to the ANOVA (row means scores) CMH statistic when the analysis uses rank scores (SCORES=RANK). The three-way table uses subject (or subject group) as the stratifying variable, treatment as the row variable, and response as the column variable. PROC FREQ handles ties by assigning midranks to tied response values. If there are multiple subjects per treatment in each block, the ANOVA CMH statistic is a generalization of Friedman s test. The data set Hypnosis contains data from a study investigating whether hypnosis has the same effect on skin potential (measured in millivolts) for four emotions (Lehmann 1975, p. 264). Eight subjects are asked to display fear, joy, sadness, and calmness under hypnosis. The data are recorded as one observation per subject for each emotion. data Hypnosis; length Emotion $ 10; input Subject Emotion $ datalines; 1 fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness fear joy sadness calmness 14.8 ; In the following PROC FREQ statements, the TABLES statement creates a three-way table stratified by Subject and a two-way table; the variables Emotion and SkinResponse form the rows and columns of each table. The CMH2 option produces the first two Cochran-Mantel-Haenszel statistics, the option SCORES=RANK specifies that rank scores are used to compute these statistics, and the NOPRINT option suppresses the contingency tables. These statements produce Output and Output proc freq data=hypnosis; tables Subject*Emotion*SkinResponse / cmh2 scores=rank noprint; run; proc freq data=hypnosis; tables Emotion*SkinResponse / cmh2 scores=rank noprint; run;

153 Example 35.10: Cochran s Q Test 2359 Because the CMH statistics in Output are based on rank scores, the Row Mean Scores Differ statistic is identical to Friedman s chi-square (Q D 6:45). The p-value of indicates that differences in skin potential response for different emotions are significant at the 10% level but not at the 5% level. When you do not stratify by subject, the Row Mean Scores Differ CMH statistic is identical to a Kruskal-Wallis test and is not significant (p= in Output ). Thus, adjusting for subject is critical to reducing the background variation due to subject differences. Output CMH Statistics: Stratifying by Subject Clinical Trial for Treatment of Pain The FREQ Procedure Summary Statistics for Emotion by SkinResponse Controlling for Subject Cochran-Mantel-Haenszel Statistics (Based on Rank Scores) Statistic Alternative Hypothesis DF Value Prob Nonzero Correlation Row Mean Scores Differ Output CMH Statistics: No Stratification Clinical Trial for Treatment of Pain The FREQ Procedure Summary Statistics for Emotion by SkinResponse Cochran-Mantel-Haenszel Statistics (Based on Rank Scores) Statistic Alternative Hypothesis DF Value Prob Nonzero Correlation Row Mean Scores Differ Example 35.10: Cochran s Q Test When a binary response is measured several times or under different conditions, Cochran s Q tests that the marginal probability of a positive response is unchanged across the times or conditions. When there are more than two response categories, you can use the CATMOD procedure to fit a repeated-measures model.

154 2360 Chapter 35: The FREQ Procedure The data set Drugs contains data for a study of three drugs to treat a chronic disease (Agresti 2002). Forty-six subjects receive drugs A, B, and C. The response to each drug is either favorable ( F ) or unfavorable ( U ). proc format; value $ResponseFmt 'F'='Favorable' 'U'='Unfavorable'; run; data drugs; input Drug_A $ Drug_B $ Drug_C $ datalines; F F F 6 U F F 2 F F U 16 U F U 4 F U F 2 U U F 6 F U U 4 U U U 6 ; The following statements create one-way frequency tables of the responses to each drug. The AGREE option produces Cochran s Q and other measures of agreement for the three-way table. These statements produce Output through Output proc freq data=drugs; tables Drug_A Drug_B Drug_C / nocum; tables Drug_A*Drug_B*Drug_C / agree noprint; format Drug_A Drug_B Drug_C $ResponseFmt.; weight Count; title 'Study of Three Drug Treatments for a Chronic Disease'; run; The one-way frequency tables in Output provide the marginal response for each drug. For drugs A and B, 61% of the subjects reported a favorable response while 35% of the subjects reported a favorable response to drug C. Output and Output display measures of agreement for the Favorable and Unfavorable levels of drug A, respectively. McNemar s test shows a strong discordance between drugs B and C when the response to drug A is favorable. Output One-Way Frequency Tables Study of Three Drug Treatments for a Chronic Disease The FREQ Procedure Drug_A Frequency Percent Favorable Unfavorable Drug_B Frequency Percent Favorable Unfavorable Drug_C Frequency Percent Favorable Unfavorable

155 Example 35.10: Cochran s Q Test 2361 Output Measures of Agreement for Drug A Favorable McNemar's Test Statistic (S) DF 1 Pr > S Simple Kappa Coefficient Kappa ASE % Lower Conf Limit % Upper Conf Limit Output Measures of Agreement for Drug A Unfavorable McNemar's Test Statistic (S) DF 1 Pr > S Simple Kappa Coefficient Kappa ASE % Lower Conf Limit % Upper Conf Limit

156 2362 Chapter 35: The FREQ Procedure Output displays the overall kappa coefficient. The small negative value of kappa indicates no agreement between drug B response and drug C response. Cochran s Q is statistically significant (p= in Output ), which leads to rejection of the hypothesis that the probability of favorable response is the same for the three drugs. Output Overall Measures of Agreement Overall Kappa Coefficient Kappa ASE % Lower Conf Limit % Upper Conf Limit Test for Equal Kappa Coefficients Chi-Square DF 1 Pr > ChiSq Output Cochran s Q Test Cochran's Q, for Drug_A by Drug_B by Drug_C Statistic (Q) DF 2 Pr > Q References Agresti, A. (1992), A Survey of Exact Inference for Contingency Tables, Statistical Science, 7(1), Agresti, A. (2002), Categorical Data Analysis, Second Edition, New York: John Wiley & Sons. Agresti, A. (2007), An Introduction to Categorical Data Analysis, Second Edition, New York: John Wiley & Sons. Agresti, A. and Coull, B. A. (1998), Approximate is Better than Exact for Interval Estimation of Binomial Proportions, The American Statistician, 52, Agresti, A., Mehta, C. R. and Patel, N. R. (1990), Exact Inference for Contingency Tables with Ordered Categories, Journal of the American Statistical Association, 85, Agresti, A. and Min, Y. (2001), On Small-Sample Confidence Intervals for Parameters in Discrete Distributions, Biometrics, 57,

157 References 2363 Agresti, A., Wackerly, D., and Boyett, J. M. (1979), Exact Conditional Tests for Cross- Classifications: Approximation of Attained Significance Levels, Psychometrika, 44, Barker, L., Rolka, H., Rolka, D., and Brown, C. (2001), Equivalence Testing for Binomial Random Variables: Which Test to Use?, The American Statistician, 55, Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, Second Edition, New York: Springer-Verlag. Birch, M. W. (1965), The Detection of Partial Association, II: The General Case, Journal of the Royal Statistical Society, B, 27, Bishop, Y., Fienberg, S. E., and Holland, P. W. (1975), Discrete Multivariate Analysis: Theory and Practice, Cambridge, MA: MIT Press. Bowker, A. H. (1948), Bowker s Test for Symmetry, Journal of the American Statistical Association, 43, Breslow, N. E. (1996), Statistics in Epidemiology: The Case-Control Study, Journal of the American Statistical Association, 91, Breslow, N. E. and Day, N. E. (1980), Statistical Methods in Cancer Research, Volume I: The Analysis of Case-Control Studies, IARC Scientific Publications, No. 32, Lyon, France: International Agency for Research on Cancer. Breslow, N. E. and Day, N. E. (1987), Statistical Methods in Cancer Research, Volume II: The Design and Analysis of Cohort Studies, IARC Scientific Publications, No. 82, Lyon, France: International Agency for Research on Cancer. Bross, I. D. J. (1958), How to Use Ridit Analysis, Biometrics, 14, Brown, L. D., Cai, T. T., and DasGupta, A. (2001), Interval Estimation for a Binomial Proportion, Statistical Science 16, Brown, M. B. and Benedetti, J. K. (1977), Sampling Behavior of Tests for Correlation in Two-Way Contingency Tables, Journal of the American Statistical Association, 72, Chow, S., Shao, J., and Wang, H. (2003), Sample Size Calculations in Clinical Research, Boca Raton, FL: CRC Press. Cicchetti, D. V. and Allison, T. (1971), A New Procedure for Assessing Reliability of Scoring EEG Sleep Recordings, American Journal of EEG Technology, 11, Clopper, C. J., and Pearson, E. S. (1934), The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial, Biometrika 26, Cochran, W. G. (1950), The Comparison of Percentages in Matched Samples, Biometrika, 37, Cochran, W. G. (1954), Some Methods for Strengthening the Common 2 Tests, Biometrics, 10,

158 2364 Chapter 35: The FREQ Procedure Collett, D. (1991), Modelling Binary Data, London: Chapman & Hall. Cohen, J. (1960), A Coefficient of Agreement for Nominal Scales, Educational and Psychological Measurement, 20, Dimitrienko, A., Molenberghs, G., Chuang-Stein, C., and Offen, W. (2005), Analysis of Clinical Trials Using SAS: A Practical Guide, Cary, NC: SAS Institute Inc. Drasgow, F. (1986), Polychoric and Polyserial Correlations in Encyclopedia of Statistical Sciences, vol. 7, ed. S. Kotz and N. L. Johnson, New York: John Wiley & Sons, Dunnett, C. W., and Gent, M. (1977), Significance Testing to Establish Equivalence Between Treatments, with Special Reference to Data in the Form of 2 2 Tables, Biometrics, 33, Farrington, C. P., and Manning, G. (1990), Test Statistics and Sample Size Formulae for Comparative Binomial Trials with Null Hypothesis of Non-zero Risk Difference or Non-unity Relative Risk, Statistics in Medicine, 9, Fienberg, S. E. (1980), The Analysis of Cross-Classified Data, Second Edition, Cambridge, MA: MIT Press. Fleiss, J. L., Levin, B., and Paik, M. C. (2003), Statistical Methods for Rates and Proportions, Third Edition, New York: John Wiley & Sons. Fleiss, J. L. and Cohen, J. (1973), The Equivalence of Weighted Kappa and the Intraclass Correlation Coefficient as Measures of Reliability, Educational and Psychological Measurement, 33, Fleiss, J. L., Cohen, J., and Everitt, B. S. (1969), Large-Sample Standard Errors of Kappa and Weighted Kappa, Psychological Bulletin, 72, Freeman, G. H. and Halton, J. H. (1951), Note on an Exact Treatment of Contingency, Goodness of Fit and Other Problems of Significance, Biometrika, 38, Gail, M. and Mantel, N. (1977), Counting the Number of r c Contingency Tables with Fixed Margins, Journal of the American Statistical Association, 72, Gail, M. and Simon, R. (1985), Tests for Qualitative Interactions between Treatment Effects and Patient Subsets, Biometrics, 41, Gart, J. J. (1971), The Comparison of Proportions: A Review of Significance Tests, Confidence Intervals and Adjustments for Stratification, Review of the International Statistical Institute, 39(2), Goodman, L. A. and Kruskal, W. H. (1979), Measures of Association for Cross Classification, New York: Springer-Verlag. Greenland, S. and Robins, J. M. (1985), Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models, Biometrics, 42, Haldane, J. B. S. (1955), The Estimation and Significance of the Logarithm of a Ratio of Frequencies, Annals of Human Genetics, 20,

159 References 2365 Hauck, W. W. and Anderson, S. (1986), A Comparison of Large-Sample Confidence Interval Methods for the Difference of Two Binomial Probabilities, The American Statistician, 40, Hirji, K. F. (2006), Exact Analysis of Discrete Data, Boca Raton, FL: Chapman & Hall/CRC. Hirji, K. F., Vollset, S. E., Reis, I. M., and Afifi, A. A. (1996), Exact Tests for Interaction in Several 2 2 Tables, Journal of Computational and Graphical Statistics, 5, Hollander, M. and Wolfe, D. A. (1999), Nonparametric Statistical Methods, Second Edition, New York: John Wiley & Sons. Jones, M. P., O Gorman, T. W., Lemka, J. H., and Woolson, R. F. (1989), A Monte Carlo Investigation of Homogeneity Tests of the Odds Ratio Under Various Sample Size Configurations, Biometrics, 45, Kendall, M. (1955), Rank Correlation Methods, Second Edition, London: Charles Griffin and Co. Kendall, M. and Stuart, A. (1979), The Advanced Theory of Statistics, vol. 2, New York: Macmillan. Kleinbaum, D. G., Kupper, L. L., and Morgenstern, H. (1982), Epidemiologic Research: Principles and Quantitative Methods, Research Methods Series, New York: Van Nostrand Reinhold. Landis, R. J., Heyman, E. R., and Koch, G. G. (1978), Average Partial Association in Three-way Contingency Tables: A Review and Discussion of Alternative Tests, International Statistical Review, 46, Leemis, L. M. and Trivedi, K. S. (1996), A Comparison of Approximate Interval Estimators for the Bernoulli Parameter, The American Statistician, 50, Lehmann, E. L. (1975), Nonparametrics: Statistical Methods Based on Ranks, San Francisco: Holden-Day. Liebetrau, A. M. (1983), Measures of Association, Quantitative Application in the Social Sciences, vol. 32, Beverly Hills: Sage Publications. Mack, G. A. and Skillings, J. H. (1980), A Friedman-Type Rank Test for Main Effects in a Two- Factor ANOVA, Journal of the American Statistical Association, 75, Mantel, N. (1963), Chi-square Tests with One Degree of Freedom: Extensions of the Mantel- Haenszel Procedure, Journal of the American Statistical Association, 58, Mantel, N. and Fleiss, J. L. (1980), Minimum Expected Cell Size Requirements for the Mantel- Haenszel One-Degree-of-Freedom Chi-Square Test and a Related Rapid Procedure, American Journal of Epidemiology, 112, Mantel, N. and Haenszel, W. (1959), Statistical Aspects of the Analysis of Data from Retrospective Studies of Disease, Journal of the National Cancer Institute, 22, Margolin, B. H. (1988), Test for Trend in Proportions, in Encyclopedia of Statistical Sciences, vol. 9, ed. S. Kotz and N. L. Johnson, New York: John Wiley & Sons,

160 2366 Chapter 35: The FREQ Procedure McNemar, Q. (1947), Note on the Sampling Error of the Difference Between Correlated Proportions or Percentages, Psychometrika, 12, Mehta, C. R. and Patel, N. R. (1983), A Network Algorithm for Performing Fisher s Exact Test in r c Contingency Tables, Journal of the American Statistical Association, 78, Mehta, C. R., Patel, N. R., and Gray, R. (1985), On Computing an Exact Confidence Interval for the Common Odds Ratio in Several 2 2 Contingency Tables, Journal of the American Statistical Association, 80, Mehta, C. R., Patel, N. R., and Senchaudhuri, P. (1991), Exact Stratified Linear Rank Tests for Binary Data, Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, ed. E.M. Keramidas, Mehta, C. R., Patel, N. R., and Tsiatis, A. A. (1984), Exact Significance Testing to Establish Treatment Equivalence with Ordered Categorical Data, Biometrics, 40, Narayanan, A. and Watts, D. (1996), Exact Methods in the NPAR1WAY Procedure, in Proceedings of the Twenty-First Annual SAS Users Group International Conference, Cary, NC: SAS Institute Inc., Newcombe, R. G. (1998), Two-sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods, Statistics in Medicine, 17, Newcombe, R. G. (1998), Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods, Statistics in Medicine, 17, Olsson, U. (1979), Maximum Likelihood Estimation of the Polychoric Correlation Coefficient, Psychometrika, 12, Pirie, W. (1983), Jonckheere Tests for Ordered Alternatives, in Encyclopedia of Statistical Sciences, vol. 4, ed. S. Kotz and N. L. Johnson, New York: John Wiley & Sons, Radlow, R. and Alf, E. F. (1975), An Alternate Multinomial Assessment of the Accuracy of the Chi-Square Test of Goodness of Fit, Journal of the American Statistical Association, 70, Robins, J. M., Breslow, N., and Greenland, S. (1986), Estimators of the Mantel-Haenszel Variance Consistent in Both Sparse Data and Large-Strata Limiting Models, Biometrics, 42, Santner, T. J. and Snell, M. K. (1980), Small-Sample Confidence Intervals for p1-p2 and p1/p2 in 2 2 Contingency Tables, Journal of the American Statistical Association, 75, Schuirmann, D. J. (1987), A Comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability, Journal of Pharmacokinetics and Biopharmaceutics, 15, Schuirmann, D. J. (1999), Confidence Interval Methods for Bioequivalence Testing with Binomial Endpoints, Proceedings of the Biopharmaceutical Section, ASA, Silvapulle, M. J. (2001), Tests Against Qualitative Interaction: Exact Critical Values and Robust Tests, Biometrics, 57,

161 References 2367 Snedecor, G. W. and Cochran, W. G. (1989), Statistical Methods, Eighth Edition, Ames: Iowa State University Press. Somers, R. H. (1962), A New Asymmetric Measure of Association for Ordinal Variables, American Sociological Review, 27, Stokes, M. E., Davis, C. S., and Koch, G. G. (2000), Categorical Data Analysis Using the SAS System, Second Edition, Cary, NC: SAS Institute Inc. Tarone, R. E. (1985), On Heterogeneity Tests Based on Efficient Scores, Biometrika, 72, 1, Theil, H. (1972), Statistical Decomposition Analysis, Amsterdam: North-Holland Publishing Company. Thomas, D. G. (1971), Algorithm AS-36. Exact Confidence Limits for the Odds Ratio in a 2 2 Table, Applied Statistics, 20, Valz, P. D. and Thompson, M. E. (1994), Exact Inference for Kendall s S and Spearman s Rho with Extensions to Fisher s Exact Test in r c Contingency Tables, Journal of Computational and Graphical Statistics, 3(4), van Elteren, P. H. (1960), On the Combination of Independent Two-Sample Tests of Wilcoxon, Bulletin of the International Statistical Institute, 37, Vollset, S. E., Hirji, K. F., and Elashoff, R. M. (1991), Fast Computation of Exact Confidence Limits for the Common Odds Ratio in a Series of 2 2 Tables, Journal of the American Statistical Association, 86, Wilson, E. B. (1927), Probable Inference, the Law of Succession, and Statistical Inference, Journal of the American Statistical Association, 22, Woolf, B. (1955), On Estimating the Relationship Between Blood Group and Disease, Annals of Human Genetics, 19, Zelen, M. (1971), The Analysis of Several 2 2 Contingency Tables, Biometrika, 58,

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163 Subject Index adjusted odds ratio, 2308 agreement, measures of, 2299 alpha level FREQ procedure, 2226, 2234 ANOVA (row mean scores) statistic, 2306 association, measures of FREQ procedure, 2270 asymmetric lambda, 2270, 2277 binomial proportion test, 2280 examples, 2345 Bowker s test of symmetry, 2299, 2300 Breslow-Day test, 2310 case-control studies odds ratio, 2294, 2308, 2309 cell count data, 2259 example (FREQ), 2335 chi-square tests examples (FREQ), 2341, 2348, 2351 FREQ procedure, 2265, 2266 Cicchetti-Allison weights, 2302 Cochran s Q test, 2299, 2303 Cochran s Q test, 2359 Cochran-Armitage test for trend, 2296, 2354 Cochran-Mantel-Haenszel statistics (FREQ), 2228, 2304, see chi-square tests ANOVA (row mean scores) statistic, 2306 correlation statistic, 2306 examples, 2353 general association statistic, 2307 cohort studies, 2353 relative risk, 2295, 2309 concordant observations, 2270 confidence limits asymptotic (FREQ), 2271 exact (FREQ), 2224 contingency coefficient, 2265, 2270 contingency tables, 2210, 2230 continuity-adjusted chi-square, 2265, 2267 correlation statistic, 2306 Cramer s V statistic, 2265, 2270 crosstabulation tables, 2210, 2230 discordant observations, 2270 exact tests computational algorithms (FREQ), 2315 computational resources (FREQ), 2317 confidence limits, 2224 FREQ procedure, 2314, 2355 network algorithm (FREQ), 2315 p-value, definitions, 2316 Fisher s exact test FREQ procedure, 2265, 2268, 2269 Fleiss-Cohen weights, 2302 Freeman-Halton test, 2269 FREQ procedure alpha level, 2226, 2234 binomial proportion, 2280, 2345 Bowker s test of symmetry, 2299 Breslow-Day test, 2310 cell count data, 2259, 2335 chi-square tests, , 2341, 2348 Cochran s Q test, 2299 Cochran-Mantel-Haenszel statistics, 2353 computation time, limiting, 2226 computational methods, 2315 computational resources, 2317, 2319 contingency coefficient, 2265 contingency table, 2348 continuity-adjusted chi-square, 2265, 2267 correlation statistic, 2306 Cramer s V statistic, 2265 default tables, 2230, 2231 displayed output, 2322 exact p-values, 2316 EXACT statement, used with TABLES, 2225 exact tests, 2224, 2314, 2355 Fisher s exact test, 2265 Friedman s chi-square statistic, 2358 gamma statistic, 2270 general association statistic, 2307 grouping variables, 2260 input data sets, 2221, 2259 kappa coefficient, 2302, 2303 Kendall s tau-b statistic, 2270 lambda asymmetric, 2270 lambda symmetric, 2270 likelihood-ratio chi-square test, 2265 Mantel-Haenszel chi-square test, 2265 McNemar s test, 2299 measures of association, 2270 missing values, 2261 Monte Carlo estimation, 2224, 2226, 2317 multiway tables, 2324, 2326, 2327

164 network algorithm, 2315 odds ratio, 2294, 2308, 2309 ODS graph names, 2334 ODS table names, 2331 one-way frequency tables, 2265, 2266, 2323, 2341 order of variables, 2222 output data sets, 2227, 2320, 2321, 2335, 2351 output variable prefixes, 2322 overall kappa coefficient, 2299 Pearson chi-square test, 2265, 2267 Pearson correlation coefficient, 2270 phi coefficient, 2265 polychoric correlation coefficient, 2270 relative risk, 2295, 2308, 2309 row mean scores statistic, 2306 scores, 2265 simple kappa coefficient, 2299 Somers D statistics, 2270 Spearman rank correlation coefficient, 2270 statistical computations, 2264 stratified table, 2353 Stuart s tau-c statistic, 2270 two-way frequency tables, 2266, 2267, 2348 uncertainty coefficients, 2270 weighted kappa coefficient, 2299 frequency tables, 2210, 2230 one-way (FREQ), 2265, 2266, 2323, 2341 two-way (FREQ), 2266, 2267, 2348 Friedman s chi-square statistic, 2358 gamma statistic, 2270, 2272 general association statistic, 2307 hypothesis tests exact (FREQ), 2224 Jonckheere-Terpstra test, 2297 kappa coefficient, 2300, 2301 tests, 2303 weights, 2302 Kendall s tau-b statistic, 2270, 2272 lambda asymmetric, 2270, 2277 lambda symmetric, 2270, 2278 likelihood-ratio chi-square test, 2265 likelihood-ratio test chi-square (FREQ), 2267 Mantel-Haenszel chi-square test, 2265, 2268 McNemar s test, 2299 measures of agreement, 2299 Mehta and Patel, network algorithm, 2315 modified ridit scores, 2265 Monte Carlo estimation FREQ procedure, 2224, 2226, 2317 network algorithm, 2315 odds ratio adjusted, 2308 Breslow-Day test, 2310 case-control studies, 2294, 2308, 2309 logit estimate, 2309 Mantel-Haenszel estimate, 2308 ODS graph names FREQ procedure, 2334 overall kappa coefficient, 2299, 2303 Pearson chi-square test, 2265, 2267 Pearson correlation coefficient, 2270, 2274 phi coefficient, 2265, 2269 polychoric correlation coefficient, 2246, 2270, 2277 prospective study, 2353 rank scores, 2265 relative risk, 2308 cohort studies, 2295 logit estimate, 2310 Mantel-Haenszel estimate, 2309 ridit scores, 2265 risks and risk differences, 2287 row mean scores statistic, 2306 simple kappa coefficient, 2299, 2300 Somers D statistics, 2270, 2274 Spearman rank correlation coefficient, 2270, 2275 stratified analysis FREQ procedure, 2210, 2230 stratified table example, 2353 Stuart s tau-c statistic, 2270, 2273 table scores, 2265 tables frequency and crosstabulation (FREQ), 2210, 2230 multiway, 2324, 2326, 2327 one-way frequency, 2323 one-way, tests, 2265, 2266 two-way, tests, 2266, 2267 Tarone s adjustment Breslow-Day test, 2310 tetrachoric correlation coefficient, 2246, 2277 trend test, 2296, 2354 uncertainty coefficients, 2270, 2279

165 weighted kappa coefficient, 2299, 2301 Yule s Q statistic, 2272 zeros, structural and random FREQ procedure, 2304

166

167 Syntax Index AGREE option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2233 TEST statement (FREQ), 2257 AJCHI option OUTPUT statement (FREQ), 2228 ALL option OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2234 ALPHA= option EXACT statement (FREQ), 2226 TABLES statement (FREQ), 2234 BDCHI option OUTPUT statement (FREQ), 2228 BDT option TABLES statement (FREQ), 2234, 2240 BINOMIAL option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2234 BINOMIALC option TABLES statement (FREQ), 2237 BY statement FREQ procedure, 2223 CELLCHI2 option TABLES statement (FREQ), 2238 CHISQ option EXACT statement (FREQ), 2225, 2348 OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2238, 2266, 2348 CL option TABLES statement (FREQ), 2239 CMH option OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2239 CMH1 option OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2240 CMH2 option OUTPUT statement (FREQ), 2228 TABLES statement (FREQ), 2240 CMHCOR option OUTPUT statement (FREQ), 2228 CMHGA option OUTPUT statement (FREQ), 2228 CMHRMS option OUTPUT statement (FREQ), 2228 COCHQ option OUTPUT statement (FREQ), 2228 COMOR option EXACT statement (FREQ), 2225 COMPRESS option PROC FREQ statement, 2221 CONTENTS= option TABLES statement (FREQ), 2241 CONTGY option OUTPUT statement (FREQ), 2228 CONVERGE= option TABLES statement (FREQ), 2241 CRAMV option OUTPUT statement (FREQ), 2228 CROSSLIST option TABLES statement (FREQ), 2241 CUMCOL option TABLES statement (FREQ), 2242 DATA= option PROC FREQ statement, 2221 DEVIATION option TABLES statement (FREQ), 2242 EQKAP option OUTPUT statement (FREQ), 2228 EQOR option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2228 EQWKP option OUTPUT statement (FREQ), 2229 EXACT option OUTPUT statement (FREQ), 2229 EXACT statement FREQ procedure, 2224 EXPECTED option TABLES statement (FREQ), 2242 FISHER option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2242 FORMAT= option TABLES statement (FREQ), 2242 FORMCHAR= option PROC FREQ statement, 2222 FREQ procedure

168 syntax, 2220 FREQ procedure, BY statement, 2223 FREQ procedure, EXACT statement, 2224 AGREE option, 2225 ALPHA= option, 2226 BINOMIAL option, 2225 CHISQ option, 2225, 2348 COMOR option, 2225 EQOR option, 2225 FISHER option, 2225 JT option, 2225 KAPPA option, 2225 KENTB option, 2225 LRCHI option, 2225 MAXTIME= option, 2226 MC option, 2226 MCNEM option, 2225 MEASURES option, 2225 MHCHI option, 2225 N= option, 2226 OR option, 2225, 2348 PCHI option, 2225 PCORR option, 2225 POINT option, 2227 RISKDIFF option, 2225 RISKDIFF1 option, 2225 RISKDIFF2 option, 2225 SCORR option, 2225 SEED= option, 2227 SMDCR option, 2225 SMDRC option, 2225 STUTC option, 2225 TREND option, 2225, 2355 WTKAP option, 2225 FREQ procedure, OUTPUT statement, 2227 AGREE option, 2228 AJCHI option, 2228 ALL option, 2228 BDCHI option, 2228 BINOMIAL option, 2228 CHISQ option, 2228 CMH option, 2228 CMH1 option, 2228 CMH2 option, 2228 CMHCOR option, 2228 CMHGA option, 2228 CMHRMS option, 2228 COCHQ option, 2228 CONTGY option, 2228 CRAMV option, 2228 EQKAP option, 2228 EQOR option, 2228 EQWKP option, 2229 EXACT option, 2229 FISHER option, 2229 GAILSIMON option, 2229 GAMMA option, 2229 JT option, 2229 KAPPA option, 2229 KENTB option, 2229 LAMCR option, 2229 LAMDAS option, 2229 LAMRC option, 2229 LGOR option, 2229 LGRRC1 option, 2229 LGRRC2 option, 2229 LRCHI option, 2229, 2351 MCNEM option, 2229 MEASURES option, 2229 MHCHI option, 2229 MHOR option, 2229 MHRRC1 option, 2229 MHRRC2 option, 2229 N option, 2229 NMISS option, 2229 OR option, 2229 OUT= option, 2227 PCHI option, 2229, 2351 PCORR option, 2229 PHI option, 2229 PLCORR option, 2229 RDIF1 option, 2229 RDIF2 option, 2229 RELRISK option, 2229 RISKDIFF option, 2229 RISKDIFF1 option, 2229 RISKDIFF2 option, 2230 RRC1 option, 2230 RRC2 option, 2230 RSK1 option, 2230 RSK11 option, 2230 RSK12 option, 2230 RSK2 option, 2230 RSK21 option, 2230 RSK22 option, 2230 SCORR option, 2230 SMDCR option, 2230 SMDRC option, 2230 STUTC option, 2230 TREND option, 2230 TSYMM option, 2230 U option, 2230 UCR option, 2230 URC option, 2230 WTKAP option, 2230 FREQ procedure, PROC FREQ statement, 2221 COMPRESS option, 2221 DATA= option, 2221

169 FORMCHAR= option, 2222 NLEVELS option, 2222 NOPRINT option, 2222 ORDER= option, 2222 PAGE option, 2223 FREQ procedure, TABLES statement, 2230 AGREE option, 2233 ALL option, 2234 ALPHA= option, 2234 BDT option, 2234, 2240 BINOMIAL option, 2234 BINOMIALC option, 2237 CELLCHI2 option, 2238 CHISQ option, 2238, 2266, 2348 CL option, 2239 CMH option, 2239 CMH1 option, 2240 CMH2 option, 2240 CONTENTS= option, 2241 CONVERGE= option, 2241 CROSSLIST option, 2241 CUMCOL option, 2242 DEVIATION option, 2242 EXPECTED option, 2242 FISHER option, 2242 FORMAT= option, 2242 GAILSIMON option, 2240, 2243 JT option, 2243 LIST option, 2243 MANTELFLEISS option, 2240 MAXITER= option, 2243 MEASURES option, 2243 MISSING option, 2244 MISSPRINT option, 2244 NOCOL option, 2244 NOCUM option, 2244 NOFREQ option, 2244 NOPERCENT option, 2244 NOPRINT option, 2245 NOROW option, 2245 NOSPARSE option, 2245 NOWARN option, 2245 OUT= option, 2245 OUTCUM option, 2245 OUTEXPECT option, 2246, 2335 OUTPCT option, 2246 PLCORR option, 2246 PLOTS= option, 2246 PRINTKWT option, 2252 RELRISK option, 2252, 2348 RISKDIFF option, 2252 RISKDIFFC option, 2255 SCORES= option, 2255, 2358 SCOROUT option, 2255 SPARSE option, 2256, 2335 TESTF= option, 2256, 2266 TESTP= option, 2256, 2266, 2342 TOTPCT option, 2256 TREND option, 2256, 2355 FREQ procedure, TEST statement, 2257 AGREE option, 2257 GAMMA option, 2257 KAPPA option, 2257 KENTB option, 2257 MEASURES option, 2257 PCORR option, 2257 SCORR option, 2257 SMDCR option, 2257, 2355 SMDRC option, 2257 STUTC option, 2257 WTKAP option, 2257 FREQ procedure, WEIGHT statement, 2258 ZEROS option, 2258 GAILSIMON option OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2240, 2243 GAMMA option OUTPUT statement (FREQ), 2229 TEST statement (FREQ), 2257 JT option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2243 KAPPA option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TEST statement (FREQ), 2257 KENTB option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TEST statement (FREQ), 2257 LAMCR option OUTPUT statement (FREQ), 2229 LAMDAS option OUTPUT statement (FREQ), 2229 LAMRC option OUTPUT statement (FREQ), 2229 LGOR option OUTPUT statement (FREQ), 2229 LGRRC1 option OUTPUT statement (FREQ), 2229 LGRRC2 option OUTPUT statement (FREQ), 2229 LIST option TABLES statement (FREQ), 2243

170 LRCHI option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229, 2351 MANTELFLEISS option TABLES statement (FREQ), 2240 MAXITER= option TABLES statement (FREQ), 2243 MAXTIME= option EXACT statement (FREQ), 2226 MC option EXACT statement (FREQ), 2226 MCNEM option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 MEASURES option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2243 TEST statement (FREQ), 2257 MHCHI option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 MHOR option OUTPUT statement (FREQ), 2229 MHRRC1 option OUTPUT statement (FREQ), 2229 MHRRC2 option OUTPUT statement (FREQ), 2229 MISSING option TABLES statement (FREQ), 2244 MISSPRINT option TABLES statement (FREQ), 2244 N option OUTPUT statement (FREQ), 2229 N= option EXACT statement (FREQ), 2226 NLEVELS option PROC FREQ statement, 2222 NMISS option OUTPUT statement (FREQ), 2229 NOCOL option TABLES statement (FREQ), 2244 NOCUM option TABLES statement (FREQ), 2244 NOFREQ option TABLES statement (FREQ), 2244 NOPERCENT option TABLES statement (FREQ), 2244 NOPRINT option PROC FREQ statement, 2222 TABLES statement (FREQ), 2245 NOROW option TABLES statement (FREQ), 2245 NOSPARSE option TABLES statement (FREQ), 2245 NOWARN option TABLES statement (FREQ), 2245 OR option EXACT statement (FREQ), 2225, 2348 OUTPUT statement (FREQ), 2229 ORDER= option PROC FREQ statement, 2222 OUT= option OUTPUT statement (FREQ), 2227 TABLES statement (FREQ), 2245 OUTCUM option TABLES statement (FREQ), 2245 OUTEXPECT option TABLES statement (FREQ), 2246, 2335 OUTPCT option TABLES statement (FREQ), 2246 OUTPUT statement FREQ procedure, 2227 PAGE option PROC FREQ statement, 2223 PCHI option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229, 2351 PCORR option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 TEST statement (FREQ), 2257 PHI option OUTPUT statement (FREQ), 2229 PLCORR option OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2246 PLOTS= option TABLES statement (FREQ), 2246 POINT option EXACT statement (FREQ), 2227 PRINTKWT option TABLES statement (FREQ), 2252 PROC FREQ statement, see FREQ procedure RDIF1 option OUTPUT statement (FREQ), 2229 RDIF2 option OUTPUT statement (FREQ), 2229 RELRISK option OUTPUT statement (FREQ), 2229 TABLES statement (FREQ), 2252, 2348 RISKDIFF option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229

171 TABLES statement (FREQ), 2252 RISKDIFF1 option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2229 RISKDIFF2 option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 RISKDIFFC option TABLES statement (FREQ), 2255 RRC1 option OUTPUT statement (FREQ), 2230 RRC2 option OUTPUT statement (FREQ), 2230 RSK1 option OUTPUT statement (FREQ), 2230 RSK11 option OUTPUT statement (FREQ), 2230 RSK12 option OUTPUT statement (FREQ), 2230 RSK2 option OUTPUT statement (FREQ), 2230 RSK21 option OUTPUT statement (FREQ), 2230 RSK22 option OUTPUT statement (FREQ), 2230 SCORES= option TABLES statement (FREQ), 2255, 2358 SCOROUT option TABLES statement (FREQ), 2255 SCORR option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 TEST statement (FREQ), 2257 SEED= option EXACT statement (FREQ), 2227 SMDCR option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 TEST statement (FREQ), 2257, 2355 SMDRC option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 TEST statement (FREQ), 2257 SPARSE option TABLES statement (FREQ), 2256, 2335 STUTC option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 TEST statement (FREQ), 2257 FREQ procedure, 2257 TESTF= option TABLES statement (FREQ), 2256, 2266 TESTP= option TABLES statement (FREQ), 2256, 2266, 2342 TOTPCT option TABLES statement (FREQ), 2256 TREND option EXACT statement (FREQ), 2225, 2355 OUTPUT statement (FREQ), 2230 TABLES statement (FREQ), 2256, 2355 TSYMM option OUTPUT statement (FREQ), 2230 U option OUTPUT statement (FREQ), 2230 UCR option OUTPUT statement (FREQ), 2230 URC option OUTPUT statement (FREQ), 2230 WEIGHT statement FREQ procedure, 2258 WTKAP option EXACT statement (FREQ), 2225 OUTPUT statement (FREQ), 2230 TEST statement (FREQ), 2257 ZEROS option WEIGHT statement (FREQ), 2258 TABLES statement FREQ procedure, 2230 TEST statement

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